Title:

Adiabatic quantum computation with superconducting qubits

Document Type and Number:

United States Patent 7418283

Abstract:

A method for quantum computing using a quantum system comprising a plurality of qubits is provided. The system can be in any one of at least two configurations at any given time including one characterized by an initialization Hamiltonian H_Oand one characterized by a problem Hamiltonian H_P. The problem Hamiltonian H_Phas a ground state. Each respective first qubit in the qubits is arranged with respect to a respective second qubit in the qubits such that they define a predetermined coupling strength. The predetermined coupling strengths between the qubits in the plurality of qubits collectively define a computational problem to be solved. In the method, the system is initialized to H_Oand is then adiabatically changed until the system is described by the ground state of the problem Hamiltonian H_P. Then the state of the system is read out by probing an observable of the σ_XPauli matrix operator.

Inventors:

Amin, Mohammad H. S. (Vancouver, CA)

Plaque It!

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims benefit, under 35 U.S.C. § 119(e), of U.S. Provisional Patent Application No. 60/557,748, filed on Mar. 29, 2004, which is hereby incorporated by reference in its entirety. This application also claims benefit, under 35 U.S.C. § 119(e), of U.S. Provisional Patent Application No. 60/588,002, filed on Jul. 13, 2004, which is hereby incorporated by reference in its entirety. This application is further related to concurrently filed application Ser. No. 11/093,205, entitled “Adiabatic Quantum Computation with Superconducting Qubits,” and application Ser. No. 11/093,201, entitled “Adiabatic Quantum Computation with Superconducting Qubits,” each of which is hereby incorporated by reference in its entirety.

Claims:

What is claimed is:

1. A method for adiabatic quantum computing using a quantum system comprising a plurality of superconducting qubits, wherein the quantum system is capable of being in any one of at least two quantum configurations at any given time, the at least two quantum configurations comprising: a first configuration described by an initialization Hamiltonian H_O; and a second configuration described by a problem Hamiltonian H_Phaving a ground state, the method comprising: A) initializing the quantum system to the first configuration; B) adiabatically changing the quantum system until it is described by the ground state of the problem Hamiltonian Hp; and C) reading out a state of the quantum system.

2. The method of claim 1 wherein each respective first superconducting qubit in the plurality of superconducting qubits is arranged with respect to a respective second superconducting qubit in the plurality of superconducting qubits such that the first respective superconducting qubit and the corresponding second respective superconducting qubit define a predetermined coupling strength and wherein the predetermined coupling strength between each first respective superconducting qubit and corresponding second respective superconducting qubit in the plurality of superconducting qubits collectively define a computational problem to be solved.

3. The method of claim 2 wherein the problem Hamiltonian H_Pcomprises a tunneling term for each of the respective superconducting qubits in the plurality of superconducting qubits, and wherein an energy of a tunneling term for each respective superconducting qubit in the plurality of superconducting qubits is less than an average of the predetermined coupling strengths between each first respective superconducting qubit and second respective superconducting qubit in the plurality of superconducting qubits.

4. The method of claim 1 wherein a superconducting qubit in the plurality of superconducting qubits is a persistent current qubit.

Description:

1. FIELD OF THE INVENTION

This invention relates to superconducting circuitry. More specifically, this invention relates to devices for quantum computation.

2. BACKGROUND

Research on what is now called quantum computing may have begun with a paper published by Richard Feynman. See Feynman, 1982 , International Journal of Theoretical Physics 21, pp. 467-488, which is hereby incorporated by reference in its entirety. Feynman noted that a quantum system is inherently difficult to simulate with conventional computers but that observation of the evolution of an analogous quantum system could provide an exponentially faster way to solve the mathematical model of the quantum system of interest. In particular, solving a mathematical model for the behavior of a quantum system commonly involves solving a differential equation related to the Hamiltonian of the quantum system. David Deutsch noted that a quantum system could be used to yield a time savings, later shown to include exponential time savings, in certain computations. If one had a problem modeled in the form of an equation that represented the Hamiltonian of a quantum system, the behavior of the system could provide information regarding the solutions to the equation. See Deutsch, 1985 , Proceedings of the Royal Society of London A 400, pp. 97-117, which is hereby incorporated by reference in its entirety.

A major activity in the quantum computing art is the identification of physical systems that can support quantum computation. This activity includes finding suitable qubits as well as developing systems and methods for controlling such qubits. As detailed in the following sections, a qubit serves as the basis for performing quantum computation.

2.1 Qubits

The physical systems that are used in quantum computing are quantum computers. A quantum bit or “qubit” is the building block of a quantum computer in the same way that a conventional binary bit is a building block of a classical computer. A qubit is a quantum bit, the counterpart in quantum computing to the binary digit or bit of classical computing. Just as a bit is the basic unit of information in a classical computer, a qubit is the basic unit of information in a quantum computer. A qubit is conventionally a system having two or more discrete energy states. The energy states of a qubit are generally referred to as the basis states of the qubit. The basis states of a qubit are termed the |0> and |1> basis states. In the mathematical modeling of these basis states, each state is associated with an eigenstate of the sigma-z (σ ^Z) Pauli matrix. See Nielsen and Chuang, 2000 , Quantum Computation and Quantum Information , Cambridge University Press, which is hereby incorporated by reference in its entirety.

The state of a qubit can be in any superposition of two basis states, making it fundamentally different from a bit in an ordinary digital computer. A superposition of basis states arises in a qubit when there is a non-zero probability that the system occupies more than one of the basis states at a given time. Qualitatively, a superposition of basis states means that the qubit can be in both basis states |0> and |1> at the same time. Mathematically, a superposition of basis states means that the wave function that characterizes the overall state of the qubit, denoted |Ψ>, has the form
|Ψ>= a| 0>+ b| 1>
where a and b are amplitudes respectively corresponding to probabilities |a| ²and |b| ². The amplitudes a and b each have real and imaginary components, which allows the phase of qubit to be modeled. The quantum nature of a qubit is largely derived from its ability to exist in a superposition of basis states, and for the state of the qubit to have a phase.

To complete a quantum computation using a qubit, the state of the qubit is typically measured (e.g., read out). When the state of the qubit is measured the quantum nature of the qubit is temporarily lost and the superposition of basis states collapses to either the |0> basis state or the |1> basis state, thus regaining its similarity to a conventional bit. The actual state of the qubit after it has collapsed depends on the amplitudes a and b immediately prior to the readout operation.

A survey of exemplary physical systems from which qubits can be formed is found in Braunstein and Lo (eds.), Scalable Quantum Computers , Wiley-VCH Verlag GmbH, Berlin (2001), which is hereby incorporated by reference in its entirety. Of the various physical systems surveyed, the systems that appear to be most suited for scaling (e.g., combined in such a manner such that they entangle with each other) are those physical systems that include superconducting structures such as superconducting qubits.

2.2 Superconducting Qubits in General

Superconducting qubits generally fall into two categories; phase qubits and charge qubits. Phase qubits store and manipulate information in the phase states of the device. Charge qubits store and manipulate information in the elementary charge states of the device. In superconducting materials, phase is a property of the material whereas elementary charges are represented by pairs of electrons called Cooper pairs. The division of such devices into two classes is outlined in Makhlin et al., 2001, “Quantum-State Engineering with Josephson-Junction Devices,” Reviews of Modern Physics 73, pp. 357-401 which is hereby incorporated by reference in its entirety.

Phase and charge are related values in superconductors and, at energy scales where quantum effects dominate, the Heisenberg uncertainty principle causes certainty in phase to lead to uncertainty in charge and, conversely, causes certainty in charge to lead to uncertainty in the phase of the system. Superconducting phase qubits are devices formed out of superconducting materials having a small number of distinct phase states and many charge states, such that when the charge of the device is certain, information stored in the phase states becomes delocalized and evolves quantum mechanically. Therefore, fixing the charge of a phase qubit leads to delocalization of the phase states of the qubit and subsequent useful quantum behavior in accordance with well-known principles of quantum mechanics.

Experimental realization of superconducting devices as qubits was made by Nakamura et al., 1999, Nature 398, p. 786, which is hereby incorporated by reference in its entirety. Nakamura et al. developed a charge qubit that demonstrates the basic operational requirements for a qubit. However, the Nakamura et al. charge qubits have unsatisfactorily short decoherence times and stringent control parameters. Decoherence time is the duration of time that it takes for a qubit to lose some of its quantum mechanical properties, e.g., the state of the qubit no longer has a definite phase. When the qubit loses it quantum mechanical properties, the phase of the qubit is no longer characterized by a superposition of basis states and the qubit is no longer capable of supporting all types of quantum computation.

Superconducting qubits have two modes of operation related to localization of the states in which information is stored. When the qubit is initialized or measured, the information is classical, 0 or 1, and the states representing that classical information are also classical in order to provide reliable state preparation. Thus, a first mode of operation of a qubit is to permit state preparation and measurement of classical information. A second mode of operation occurs during quantum computation, where the information states of the device become dominated by quantum effects such that the qubit can evolve controllably as a coherent superposition of those states and, in some instances, even become entangled with other qubits in the quantum computer. Thus, qubit devices provide a mechanism to localize the information states for initialization and readout operations, and de-localize the information states during computation. Efficient functionality of both of these modes and, in particular, the transition between them in superconducting qubits is a challenge that has not been satisfactorily resolved in the prior art.

2.2.1 Phase Qubits

A proposal to build a quantum computer from superconducting qubits was published in 1997. See Bocko et al., 1997, IEEE Trans. Appl. Supercon. 7, p. 3638, which is hereby incorporated by reference in its entirety. See also Makhlin et al., 2001, Rev. Mod. Phys. 73, p. 357 which is hereby incorporated by reference in its entirety. Since then, designs based on many other types of qubits have been introduced. One such design is based on the use of superconducting phase qubits. See Mooij et al., 1999, Science 285, 1036; and Orlando et al., 1999, Phys. Rev . B 60, 15398, which are hereby incorporated by reference in their entireties. In particular, quantum computers based on persistent current qubits, which are one type of superconducting phase qubit, have been proposed.

The superconducting phase qubit is well known and has demonstrated long coherence times. See, for example, Orlando et al., 1999, Phys. Rev. B 60, 15398, and Il'ichev et al., 2003, Phys. Rev. Lett. 91, 097906, which are hereby incorporated by reference in their entireties. Some other types of superconducting phase qubits comprise superconducting loops interrupted by more or less than three Josephson junctions. See, e.g., Blatter et al., 2001, Phys. Rev. B 63, 174511, and Friedman et al., 2000, Nature, 406, 43, which are hereby incorporated by reference in their entireties.

FIG. 1A illustrates a persistent current qubit 101 . Persistent current qubit 101 comprises a loop 103 of superconducting material interrupted by Josephson junctions 101 - 1 , 101 - 2 , and 101 - 3 . Josephson junctions are typically formed using standard fabrication processes, generally involving material deposition and lithography stages. See, e.g., Madou, Fundamentals of Microfabrication, Second Edition , CRC Press, 2002, which is hereby incorporated by reference in its entirety. Methods for fabricating Josephson junctions are well known and described in Ramos et al., 2001, IEEE Trans. App. Supercond. 11, 998, for example, which is hereby incorporated by reference in its entirety. Details specific to persistent current qubits can be found in C. H. van der Wal, 2001; J. B. Majer, 2002; and J. R. Butcher, 2002, all Theses in Faculty of Applied Sciences, Delft University of Technology, Delft, The Netherlands; http://qt.tn.tudelft.nl; Kavli Institute of Nanoscience Delft, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands, which is hereby incorporated by reference in its entirety. Common substrates include silicon, silicon oxide, or sapphire, for example. Josephson junctions can also include insulating materials such as aluminum oxide, for example. Exemplary superconducting materials useful for forming superconducting loop 103 are aluminum and niobium. The Josephson junctions have cross-sectional sizes ranging from about 10 nanometers (nm) to about 10 micrometers (μm). One or more of the Josephson junctions 101 has parameters, such as the size of the junction, the junction surface area, the Josephson energy or the charging energy that differ from the other Josephson junctions in the qubit.

The difference between any two Josephson junctions in the persistent current qubit is characterized by a coefficient, termed α, which typically ranges from about 0.5 to about 1.3. In some instances, the term a for a pair of Josephson junctions in the persistent current qubit is the ratio of the critical current between the two Josephson junctions in the pair. The critical current of a Josephson junction is the minimum current through the junction at which the junction is no longer superconducting. That is, below the critical current, the junction is superconducting whereas above the critical current, the junction is not superconducting. Thus, for example, the term a for junctions 101 - 1 and 101 - 2 is defined as the ratio between the critical current of junction 101 - 1 and the critical current of junction 101 - 2 .

Referring to FIG. 1A, a bias source 110 is inductively coupled to persistent current qubit 101 . Bias source 110 is used to thread a magnetic flux Φ _xthrough phase qubit 101 to provide control of the state of the phase qubit. In some instances, the persistent current qubit operates with a magnetic flux bias Φ _xranging from about 0.2·Φ ₀to about 0.8·Φ ₀, where Φ ₀is the flux quantum. In some instances, the magnetic flux bias ranges from about 0.47·Φ ₀to about 0.5·Φ ₀.

Persistent current qubit 101 has a two-dimensional potential with respect to the phase across Josephson junctions 101 - 1 , 101 - 2 , and 101 - 3 . In some instances, persistent current qubit 101 is biased with a magnetic flux Φ _x, such that the two-dimensional potential profile includes regions of local energy minima, where the local energy minima are separated from each other by small energy barriers and are separated from other regions by large energy barriers. In some instances, this potential has the shape of double well potential 100 B (FIG. 1B), which includes a left well 160 - 0 and a right well 160 - 1 . In such instances, left well 160 - 0 can represent clockwise ( 102 - 0 ) circulating supercurrent in the phase qubit 101 and right well 160 - 1 can represent counter-clockwise ( 102 - 1 ) circulating supercurrent in persistent current qubit 101 of FIG. 1A.

When wells 160 - 0 and 160 - 1 are at or near degeneracy, meaning that they are at the same or nearly the same energy potential as illustrated in FIG. 1B, the quantum state of persistent current qubit 101 becomes a coherent superposition of the phase or basis states and device can be operated as a phase qubit. The point at or near degeneracy is herein referred to as the point of computational operation of the persistent current. During computational operation of the persistent current qubit, the charge of the qubit is fixed leading to uncertainty in the phase basis and delocalization of the phase states of the qubit. Controllable quantum effects can then be used to process the information stored in those phase states according to the rules of quantum mechanics. This makes the persistent current qubit robust against charge noise and thereby prolongs the time under which the qubit can be maintained in a coherent superposition of basis states.

2.2.2 Charge Qubits

There are broad classes of condensed matter systems that have states defined by the presence and absence of extra charge, or the excess charge exists in either a ground or an excited state. Such systems are diverse and have long held theoretical and experimental interest, e.g. Millikan, 1911, Phys. Rev. 32, pp. 349-397, which is hereby incorporated by reference in its entirety. There has been attention directed to semiconductor systems such as quantum dots. Research has been conducted using single particle electronics because they hold promise for conventional computers. Subsequently, proposals were made for these systems as quantum computers. While these systems collectively could be called charge qubits, this term is reserved herein for superconducting qubits. Specifically, a superconducting charge qubit has as basis states the presence (charge=2e, or some multiple thereof of 2e) or absence (charge=0) of charge on a small superconducting island. For charge qubits, the Coulomb energy E _C=e ²/2C exceeds the Josephson energy E _Jof the qubit.

A charge qubit is a small (mesoscopic) island of superconductor separated by a Josephson junction from a large superconductor (reservoir), see FIG. 12, for example. This system can be tuned to behave like an ideal two-level quantum system. Classical basis states |0> and |1> (corresponding to the presence or absence of a Cooper pair) are the working states of the charge qubit. The Hamiltonian of the superconducting charge qubit is,

$H = - \frac{1}{2} [4 E_{C} (1 - 2 n_{g}) σ^{Z} + E_{J} σ^{X}]$
where the dimensionless gate charge n _g≈C _gV _g/2e is determined in operation by the gate voltage V _g, and in fabrication by the capacitance C _g. Here it is assumed that n _g≈½. The bias term for the charge qubit is proportional to σ ^Z. A finite Josephson energy allows transition between the states with tunnel splitting Δ proportional to E _J. The Josephson energy of the Josephson junction connecting the island to the reservoir can be made tunable. Tuning the dimensionless gate and the Josephson energy allows one to independently bias the charge and vary the tunneling rate of the charge qubit. See Nakamura et al., 1999, Nature 398, pp. 786-788; and Makhlin et al., 2001, Rev. Mod. Phys. 73, pp. 357-401, each of which is hereby incorporated by reference in its entirety.

2.3 NP Complexity Classes

Computer scientists concerned with complexity routinely use the definitions of different complexity classes. The number of complexity classes is ever changing, as new ones are defined and existing ones merge through advancements made in computer science. The complexity classes known as non-deterministic polynomial-time (NP), NP-complete (NPC), and NP-hard (NPH) are all classes of decision problems. Decision problems have binary outcomes.

Problems in NP are computational problems for which there exists polynomial time verification. That is, it takes no more than polynomial time (class P) in the size of the problem to verify a potential solution. It may take more than polynomial time to create a potential solution. NP-hard problems take longer to verify a potential solution. For each NP-hard problem, there is an NP-complete problem that can be reduced to the NP-hard problem. However, NP-complete problems that can be reduced to a NP-hard problem do not enjoy polynomial time verification.

Problems in NPC can be defined as problems in NP that have been shown to be equivalent to, or harder to solve, than a known problem in NPC. Equivalently, the problems in NPC are problems in NP that are also in NPH. This can be expressed as NPC=NP∩NPH.

A problem is equivalent, or harder to solve, than a known problem in NPC if there exists a polynomial time reduction to the instant problem from the known problem in NPC. Reduction can be regarded as a generalization of mapping. The mappings can be one to one functions, many to one functions, or make use of oracles, etc. The concepts of complexity classes and how they define the intractability of certain computational problems is found in, for example, Garey and Johnson, 1979, Computers and Intractability: A Guide to the Theory of NP - Completeness , Freeman, San Francisco, ISBN: 0716710455, which is hereby incorporated by reference in its entirety. Also see, Cormen, Leiserson, and Rivest, 1990 , Introduction to Algorithms , MIT Press, Cambridge, ISBN: 0262530910.

2.4 Circuit Model of Quantum Computing

Analogous to the way a classical computer is built using wires and logic gates, a quantum computer can be built using quantum circuits comprised of “wires” and “unitary gates.” Here, the wire is not a physical entity. Rather, it represents the state of the qubit in time. The “unitary gates” are applied at precise times to specific qubits to effect evolution of the qubit in accordance with the circuit model for quantum computing. The circuit model of quantum computing is a standard and universal model used by many practitioners in the art. The circuit model is universal in the sense that it is able to convert any input state into any output state. The elements of the circuit model are that a small set of one- and two-qubit unitary gates are applied to the qubits with precise timing. The circuit model of quantum computing can implement algorithms such as Shor's algorithm for factoring numbers or Grover's algorithm for searching databases. Shor's algorithm provides an exponential speedup relative to classical (non-quantum) computers for factoring numbers. Grover's application provides a polynomial speed up relative to classical computers for searching databases. See, for example, Nielsen and Chuang, 2000, Quantum Computation and Quantum Information , Cambridge University Press, which is hereby incorporated by reference in its entirety.

An example of the circuit model is shown in FIG. 2. Circuit 200 is an implementation of the quantum Fourier transform. The quantum Fourier transform is a useful procedure found in many quantum computing applications based on the circuit model. See, for example, United States Patent Publication 2003/0164490 A1, entitled “Optimization process for quantum computing process,” which is hereby incorporated by reference in its entirety. Time progresses from left to right, i.e., time step 201 precedes time step 202 , and so forth. The four qubits in the quantum system described by FIG. 2 are indexed 0 - 3 from bottom to top. The state of qubit 0 at any given time step is represented by wire S 0 -S 0 ′, the state of qubit 1 at any give time step is represented by S 1 -S 1 ′, etc. In time step 201 , a single-qubit unitary gate, A ₃, is applied to qubit 3 . The next gate on wire S 3 -S 3 ′ for qubit 3 is a two-qubit gate, B ₂₃, which is applied to qubits 2 and 3 at time step 202 . In general the A _igate (e.g., A ₃as applied to qubit 3 at time step 201 ) is a H ADAMARD gate applied on the i ^thqubit while the B _ijgate (e.g., B ₂₃which is applied to qubits 2 and 3 at time step 202 ) is a CPHASE gate coupling the i ^thand j ^thqubit. The application of unitary gates continues until states S 0 -S 3 have been converted to S 0 ′-S 3 ′. After time step 210 , more unitary gates can be applied to the qubits or the states of the qubits can be determined (e.g., by measurement).

2.5 Adiabatic Model of Quantum Computation

The following subsections discuss the adiabatic theorem of quantum mechanics and introduce adiabatic quantum computing.

2.5.1 Adiabatic Theorem of Quantum Mechanics

One definition of an adiabatic process is a process that occurs in a system without heat entering or leaving the system. There exists a theorem in quantum mechanics that provides a suitable framework for such processes. The adiabatic theorem of quantum mechanics has several versions but a notable element of many such versions is as follows. A quantum system prepared in its ground state will remain in the ground state of the various instantaneous Hamiltonians through which it passes, provided the changes are made sufficiently slowly. This form of change is termed adiabatic change. Such a system is adiabatic because the population of the various states of the quantum system has not been altered as a result of the change. Hence, if the populations have not changed, the temperature of the system has not changed, and therefore no heat has entered or left the system.

2.5.2 Adiabatic Quantum Computing

In 2000, a form of quantum computing, termed adiabatic quantum computing, was proposed. See, for example, Farhi et al., 2001, Science 292, pp. 472-475, which is hereby incorporated by reference in its entirety. In adiabatic quantum computing (AQC), the problem to be solved is encoded into a physical system such that departures from the solution to the problem incur a net energy cost to the system. AQC is universal in that it is able to convert any input state into any output state. However, unlike the circuit model of quantum computing, there is no application of a predetermined set of one- and two-qubit unitary gates at precise times. It is believed that AQC can be used to find solutions to some problems with greater efficiency than the circuit model. Such problems include problems contained in, and related to, the NP, NP-hard, and NP-complete classes.

As shown in FIG. 3, AQC involves initializing a system, which encodes a problem to be solved, to an initial state. This initial state is described by an initial Hamiltonian H ₀. Then the system is migrated adiabatically to a final state described by Hamiltonian H _P. The final state encodes a solution to the problem. The migration from H ₀to H _Pfollows an interpolation path described by function γ(t) that is continuous over the time interval zero to T, inclusive, and has a condition that the magnitude of initial Hamiltonian H ₀is reduced to zero after time T. Here, T, refers to the time point at which the system reaches the state represented by the Hamiltonian H _P. Optionally, the interpolation can traverse an extra Hamiltonian H _Ethat can contain tunneling terms for some or all of the qubits represented by H ₀. The magnitude of extra Hamiltonian H _Eis described by a function δ(t) that is continuous over the time interval zero to T, inclusive, and is zero at the start (t=0) and end (t=T) of the interpolation while being non-zero at all or a portion of the times between t=0 and t=T.

One computational problem that can be solved with adiabatic quantum computing is the MAXCUT problem. Consider an undirected edge-weighted graph having a set of vertices and a set of edges. All the edges in the graph have weights given by a positive integer. The MAXCUT problem, expressed as a decision problem, asks whether there is a partition of the graph such that the sum of the weights of the edges crossing the partition is equal or greater than some given predefined positive integer K. Many other permutations of the problem exist and include optimization problems based on this decision problem. An example of an optimization problem is the identification of the partition of the graph that yields the maximum K. In other words, for graph G=(V, E) that is a (not necessarily simple) undirected edge-weighted graph with nonnegative weights, where a cut C of G is any nontrivial subset of V, the weight of cut C is the sum of weights of edges crossing the cut. The MAXCUT problem, expressed as an optimization problem, is the identification of a cut G having the maximum possible weight.

The MAXCUT problem, expressed as a decision problem, is defined in Garey and Johnson, 1979 , Computers and Intractability: A Guide to the Theory of NP - Completeness , Freeman, San Francisco, ISBN: 0716710455, which is hereby incorporated by reference in its entirety, as:

- INSTANCE: Graph G=(V, E), weight w(e)εZ ⁺ for each eεE, for positive integer K.
- QUESTION: Is there a partition of V into disjoint sets V ₁and V ₂such that the sum of the weights of the edges form E that have one endpoint in V ₁and one endpoint in V ₂is at least K?

Consider an instance of a positive number K and a graph G=(V, E), having a set of vertices V={v ₁, . . . , v _|V|}, and a set of edges E={e ₁, . . . , e _i, . . . , e _|E|}, where e _i=(v _j,v _k) for all 1<j,k<|V|. The graph's edges have weights w(e _i), w(v _j,v _k), or w _jkthat are positive. The explicit decision problem is whether there is a partition of V, i.e., V ₁ ⊂ V, V ₂ ⊂ V, and V ₁∪V ₂=V, such that the sum of the weights of the edges crossing the partition is equal or greater than some given predefined positive integer K, e.g.,

$\sum_{\forall v_{j} {εV}_{1}} \sum_{\forall v_{k} {εV}_{2}} w (v_{j}, v_{k}) ≧ K .$
An optional addition to the definitions above is the graph may have vertex weights that are also positive w(v _i) or w _i. Using this alternative, MAXCUT can be formulated as a search for a partition of G such that the sum of the weights of the edges crossing the partition, and the sum of the weight of vertices on one side of the partition is equal or greater than K. M AXCUT is a problem that has been solved using a nuclear magnetic resonance (NMR) quantum computer. See, for example, M. Steffen, Wim van Dam, T. Hogg, G. Breyta, and I. Chuang, 2003, “Experimental Implementation of an Adiabatic Quantum Optimization Algorithm,” Phys. Rev. Lett. 90, 067903, which is hereby incorporated by reference in its entirety.

Mathematically, solving MAXCUT permits optimizations based on MAXCUT to be solved efficiently. In other words, efficiency in solving a decision-based MAXCUT problem (e.g., is there a cut having a value greater than some predetermined given value K) will lead to efficiency in solving the corresponding optimization-based MAXCUT problem (finding the cut having the greatest value). This is generally true of any problem in NP. However, for problems in NPH, their related optimization problems represent a class for which adiabatic quantum computing can be particularly well suited.

One computational problem that can be solved with adiabatic quantum computing is the INDEPENDENT SET problem. Garey and Johnston, 1979 , Computers and Intractability: A Guide to the Theory of NP - Completeness , define the INDEPENDENT SET problem as:

- INSTANCE: Graph G=(V, E), positive integer K≦|V|.
- QUESTION: Does G contain an independent set of size K or more, i.e., as subset of V′ ⊂ V with |V′|≧K such that no two vertices in V′ are joined by an edge in E?
  where emphasis is added to show differences between the INDEPENDENT SET problem and another problem, known as CLIQUE , that is described below. Expanding upon this definition, consider an undirected edge-weighted graph having a set of vertices and a set of edges, and a positive integer K that is less than or equal to the number of vertices of the graph. The INDEPENDENT SET problem, expressed as a decision problem, asks whether there is a subset of vertices of size K, such that no two vertices in the subset are connected by an edge of the graph. Many other permutations of the problem exist and include optimization problems based on this decision problem. An example of an optimization problem is the identification of the independent set of the graph that yields the maximum K. This is called MAX INDEPENDENT SET.

Mathematically, solving INDEPENDENT SET permits optimizations based on INDEPENDENT SET , such as MAX INDEPENDENT SET to be solved efficiently. In other words, efficiency in solving a decision-based INDEPENDENT SET problem (e.g., is there an independent set having a value greater that some predetermined given value K) will lead to efficiency in solving the corresponding optimization-based INDEPENDENT SET problem (finding the independent set having the greatest value). This is generally true of any problem in NP.

Mathematically, solving INDEPENDENT SET permits the solving of yet another problem known as CLIQUE . This problem seeks the clique in a graph. A clique is a set of vertices that are all connected to each other. Given a graph, and a positive integer K, the question that is asked in CLIQUE is whether there are K vertices all of which are neighbors of each other. Like the INDEPENDENT SET problem, the CLIQUE problem can be converted to an optimization problem. The computation of cliques has roles in economics and cryptography. Solving an independent set on graph G ₁=(V, E) is equivalent to solving clique on G ₁'s complement G ₂=(V,(V×V)/E), e.g., for all vertices connected by edges in E remove the edges, insert into G ₂edges connecting vertices not connected in G ₁. Garey and Johnston define CLIQUE as:

- INSTANCE: Graph G=(V, E), positive integer K≦|V|.
- QUESTION: Does G contain a clique of size K or more, i.e., as subset of V′ ⊂ V with |V′|≧K such that every two vertices in V′ are joined by an edge in E?
  Here, emphasis has been added to show differences between CLIQUE and INDEPENDENT SET . It can also be shown how CLIQUE is related to the problem VERTEX COVER . Again, all problems in NP-complete are reducible to each other within polynomial time, making devices that solve one NP-complete problem efficiently, useful for other NP-complete problems.

2.6 Adiabatic Quantum Computing Using Superconducting Qubits

The question of whether superconducting qubits can be used to implement adiabatic quantum computing (AQC) has been posed in the art. However, such proposals are unsatisfactory because they either lack enabling details on the physical systems on which AQC would be implemented or they rely on qubits that have not been shown to successfully perform an n-qubit quantum computation, where n is greater than 1 and the quantum computation requires entanglement of qubits. For example, Kaminsky and Lloyd, 2002, “Scalable Architecture for Adiabatic Quantum Computing of NP-Hard Problems,” in Quantum Computing & Quantum Bits in Mesoscopic Systems , Kluwer Academic, Dordrecht, Netherlands, also published as arXiv.org: quant-ph/0211152, which is hereby incorporated by reference in its entirety, suggests that AQC can be performed with persistent current qubits, without explicitly stating how. As another example, W. M. Kaminsky, S. Lloyd, T. P. Orlando, 2004, “Scalable Superconducting Architecture for Adiabatic Quantum Computation,” arXiv.org: quant-ph/0403090, hereby incorporated by reference, describes a method and structure for AQC for a type of persistent current qubit that has not been shown to support multi-qubit quantum computation. This reference shows a system of logical qubits without giving explicit construction of the logical qubits from physical qubits or an explicit coupling between more than two qubits.

Accordingly, given the above background, there is a need in the art for improved systems and methods for adiabatic quantum computing. Discussion or citation of a reference herein shall not be construed as an admission that such reference is prior art to the present invention.

3. SUMMARY OF THE INVENTION

The present invention addresses the need in the art for improved systems and methods for adiabatic quantum computing. In some embodiments of the present invention, a graph based computing problem, such as MAXCUT , is represented by an undirected edge-weighted graph. Each node in the edge-weighted graph corresponds to a qubit in a plurality of qubits. The edge weights of the graph are represented in the plurality of qubits by the values of the coupling energies between the qubits. For example, the edge weight between a first and second node in the graph is represented by the coupling energy between a corresponding first and second qubit in the plurality of qubits.

In one aspect of the present invention, the plurality of qubits that represents the graph is initialized to a first state that does not permit the qubits to quantum tunnel. Then, the plurality of qubits is set to an intermediate state in which quantum tunneling between individual basis states within each qubit in the plurality of qubits can occur. In preferred embodiments, the change to the intermediate state occurs adiabatically. In other words, for any given instant t that occurs during the change to the intermediate state or while the qubits are in the intermediate state, the plurality of qubits are in the ground state of an instantaneous Hamiltonian that describes the plurality of qubits at the instant t. The qubits remain in the intermediate state that permits quantum tunneling between basis states for a period of time that is sufficiently long enough to allow the plurality of qubits to reach a solution for the computation problem represented by the plurality of qubits.

Once the qubits have been permitted to quantum tunnel for a sufficient period of time, the state of the qubits is adjusted such that they reach some final state that either does not permit quantum tunneling or, at least, does not permit rapid quantum tunneling. In preferred embodiments, the change to the final state occurs adiabatically. In other words, for any given instant t that occurs during the change to the final state, the plurality of qubits are in the ground state of an instantaneous Hamiltonian that describes the plurality of qubits at the instant t.

In other examples of the systems and methods of the present invention, the plurality of qubits that represents the graph is initialized to a first state that does permit the qubits to quantum tunnel. The state of the quantum system is changed once the qubits have been permitted to quantum tunnel for a sufficient period of time. The state of the qubits is adjusted such that they reach some final state that either does not permit quantum tunneling or, at least, does not permit rapid quantum tunneling. In preferred embodiments, the change to the final state occurs adiabatically.

Some embodiments of the present invention are universal quantum computers in the adiabatic quantum computing model. Some embodiments of the present invention include qubits with single-qubit Hamiltonian terms and at least one two-qubit Hamiltonian term.

A first aspect of the invention provides a method for quantum computing using a quantum system comprising a plurality of superconducting qubits. The quantum system is characterized by an impedance. Also, the quantum system is capable of being in any one of at least two configurations at any given time. These at least two configurations include a first configuration characterized by an initialization Hamiltonian H _Oas well as a second configuration characterized by a problem Hamiltonian H _P. The problem Hamiltonian has a ground state. Each respective first superconducting qubit in the plurality of superconducting qubits is arranged with respect to a respective second superconducting qubit in the plurality of superconducting qubits such that the first respective superconducting qubit and the corresponding second respective superconducting qubit define a predetermined coupling strength. The predetermined coupling strengths between each of the first respective superconducting qubit and corresponding second respective superconducting qubit collectively define a computational problem to be solved. In this first aspect of the invention, the method comprises initializing the quantum system to the initialization Hamiltonian H _O. The quantum system is then adiabatically changed until it is described by the ground state of the problem Hamiltonian H _P. The state of the quantum system is then read out by probing an observable of the σ _XPauli matrix operator.

In some embodiments in accordance with the first aspect of the invention, the reading step comprises measuring an impedance of the quantum system. In some embodiments the reading step comprises determining a state of a superconducting qubit in the plurality of superconducting qubits. In some embodiments, the reading step differentiates a ground state of the superconducting qubit from an excited state of the superconducting qubit. In some embodiments, a superconducting qubit in the plurality of superconducting qubits is a persistent current qubit. In some embodiments, the reading step measures a quantum state of the superconducting qubit as a presence or an absence of a voltage. In some embodiments, a superconducting qubit in the plurality of superconducting qubits is capable of tunneling between a first stable state and a second stable state when the quantum system is in the first configuration.

In some embodiments, a superconducting qubit in the plurality of superconducting qubits is capable of tunneling between a first stable state and a second stable state during the adiabatic changing step. In some embodiments, the adiabatic changing step occurs during a time period that is between 1 nanosecond and 100 microseconds. In some embodiments, the initializing step includes applying a magnetic field to the plurality of superconducting qubits in the direction of a vector that is perpendicular to a plane defined by the plurality of superconducting qubits. In some embodiments, a superconducting qubit in the plurality of superconducting qubits is a persistent current qubit.

A second aspect of the invention provides a method for quantum computing using a quantum system that comprises a plurality of superconducting qubits. The quantum system is coupled to an impedance readout device. The quantum system is capable of being in any one of at least two configurations at any given time. The at least two configurations include a first configuration characterized by an initialization Hamiltonian H ₀, and a second Hamiltonian characterized by a problem Hamiltonian H _P. The problem Hamiltonian H _Phas a ground state. Each respective first superconducting qubit in the plurality of superconducting qubits is arranged with respect to a respective second superconducting qubit in the plurality of superconducting qubits such that the first respective superconducting qubit and the second respective superconducting qubit define a predetermined coupling strength. The predetermined coupling strength between each said first respective superconducting qubit and corresponding second respective superconducting qubit collectively define a computational problem to be solved. In this second aspect of the invention, method comprises initializing the quantum system to the initialization Hamiltonian H _O. Then the quantum system is adiabatically changed until it is described by the ground state of the problem Hamiltonian H _P. The state of the quantum system is then read out through the impedance readout device thereby solving the computational problem.

In some embodiments in accordance with this second aspect of the invention, the reading step measures a quantum state of a superconducting qubit in the plurality of superconducting qubits as a presence or an absence of a voltage. In some embodiments, the reading step differentiates a ground state of the superconducting qubit from an excited state of the superconducting qubit. In some embodiments, a superconducting qubit in the plurality of superconducting qubits is (i) a phase qubit in the charge regime or (ii) a persistent current qubit. In some embodiments, a superconducting qubit in the plurality of superconducting qubits is capable of tunneling between a first stable state and a second stable state when the quantum system is in the first configuration. In some embodiments, a superconducting qubit in the plurality of superconducting qubits is capable of tunneling between a first stable state and a second stable state during the adiabatic changing step. In some embodiments, the adiabatic changing step occurs during a time period that is greater than 1 nanosecond and less than 100 microseconds. In some embodiments, the initializing step includes applying a magnetic field to the plurality of superconducting qubits in the direction of a vector that is perpendicular to a plane defined by the plurality of superconducting qubits. In some embodiments, a superconducting qubit in the plurality of superconducting qubits is a persistent current qubit.

A third aspect of the invention provides a method of determining a quantum state of a first target superconducting qubit. The method comprises presenting a plurality of superconducting qubits including a first target superconducting qubit in the plurality of superconducting qubits. A problem Hamiltonian describes (i) the quantum state of the plurality of superconducting qubits and (ii) each coupling energy between qubits in the plurality of qubits. The problem Hamiltonian is at or near a ground state. An rf-flux is added to the first target superconducting qubit. The rf-flux has an amplitude that is less than one flux quantum. An amount of an additional flux in the first target superconducting qubit is adiabatically varied. A presence or an absence of a dip in a voltage response of a tank circuit that is inductively coupled with the first target superconducting qubit during the adiabatically varying step is observed thereby determining the quantum state of the first target superconducting qubit.

In some embodiments in accordance with this third aspect of the invention, each superconducting qubit in the plurality of superconducting qubits is in a quantum ground state during all or a portion of the adding step, the adiabatically varying step, and the observing step. In some embodiments, the problem Hamiltonian corresponds to a terminus of an adiabatic evolution of the plurality of superconducting qubits. In some embodiments, the method further comprises biasing all or a portion of the superconducting qubits in the plurality of superconducting qubits. The problem Hamiltonian further describes a biasing on the first target superconducting qubit. In some embodiments, an energy of the biasing step exceeds the tunneling energy of a tunneling element of the Hamiltonian of the first target superconducting qubit, thereby causing tunneling to be suppressed in the first target superconducting qubit during an instance of the biasing step, adding step and the adiabatically varying step.

In some embodiments in accordance with this third aspect of the invention, the method further comprises adiabatically removing additional flux that was added to the first target superconducting qubit during the adiabatically varying step. In some embodiments, the adiabatically varying step comprises adiabatically varying the additional flux in accordance with a waveform selected from the group consisting of periodic, sinusoidal, triangular, and trapezoidal. In some embodiments, the adiabatically varying step comprises adiabatically varying the additional flux in accordance with a low harmonic Fourier approximation of a waveform selected from the group consisting of periodic, sinusoidal, triangular, and trapezoidal. In some embodiments, the additional flux has a direction that is deemed positive or negative. In some embodiments, the adiabatically varying step is characterized by a waveform that has an amplitude that grows with time. The amplitude of the waveform corresponds to an amount of additional flux that is added to the first target superconducting qubit during the adiabatically varying step. In some embodiments, the additional flux has an equilibrium point that varies with time. In some embodiments, the additional flux is either unidirectional or bidirectional. In some embodiments, the additional flux has a frequency of oscillation between about 1 cycle per second and about 100 kilocycles per second.

In some embodiments in accordance with the third aspect of the invention, the adding step comprises adding the rf-flux using (i) an excitation device that is inductively coupled to the first target superconducting qubit or (ii) a the tank circuit. In some embodiments, the method further comprises repeating the adding step and the adiabatically varying step between 1 time and 100 times. In such embodiments, the presence or absence of the dip in the voltage response of the tank circuit is observed as an average of the voltage response of the tank circuit across each instance of the adiabatically varying step.

In some embodiments in accordance with the third aspect of the invention, the first target superconducting qubit is flipped from an original basis state to an alternate basis state during the adiabatically varying step. The method further comprises returning the first target superconducting qubit to its original basis state by adiabatically removing additional flux in the qubit after the adiabatically varying step. In some embodiments, the adiabatically varying step does not alter the quantum state of each of superconducting qubits in the plurality of superconducting qubits other than the first target superconducting qubit. In some embodiments, the method further comprises recording a presence or an absence of the dip in the voltage response of the tank circuit.

In some embodiments in accordance with the third aspect of the invention, the method further comprises adding a second rf-flux to a second target superconducting qubit in the plurality of superconducting qubits. The second rf-flux has an amplitude that is less than one flux quantum. Then an amount of a second additional flux in the second target superconducting qubit is adiabatically varied. A presence or an absence of a second dip in a voltage response of a tank circuit that is inductively coupled with the second target superconducting qubit during said adiabatically varying is observed, thereby determining the quantum state of the second target superconducting qubit.

In some embodiments in accordance with the third aspect of the invention, the method further comprises designating a different superconducting qubit in the plurality of superconducting qubits as the first target superconducting qubit. The adding step and the adiabatically varying step are then reperformed with the different superconducting qubit as the first target superconducting qubit. The designating and reperforming are repeated until all or a portion (e.g., most, almost all, at least eighty percent) of the superconducting qubits in the plurality of superconducting qubits have been designated as the first target superconducting qubit.

In some embodiments in accordance with the third aspect of the invention, a tank circuit is inductively coupled with the first target superconducting qubit. The method further comprises performing an adiabatic quantum computation step for an amount of time with the plurality of superconducting qubits prior to the adding step. The amount of time is determined by a factor the magnitude of which is a function of a number of qubits in the plurality of superconducting qubits. An amount of an additional flux in the first target superconducting qubit is adiabatically varied. Then, a presence or an absence of a dip in the voltage response of a tank circuit during the adiabatically varying step is observed, thereby determining the quantum state of the first target superconducting qubit. In some embodiments, the presence of a dip in the voltage response of the tank circuit corresponds to the first target superconducting qubit being in a first basis state. The absence of a dip in the voltage response of the tank circuit corresponds to the target superconducting qubit being in a second basis state.

In some embodiments in accordance with the third aspect of the invention, the adiabatically varying step further comprises identifying an equilibrium point for the additional flux using an approximate evaluation method. In some embodiments, the method further comprises classifying the state of the first target qubit as being in the first basis state when the dip in the voltage across the tank circuit occurs to the left of the equilibrium point and classifying the state of the first target qubit as being in the second basis state when the dip in the voltage across the tank circuit occurs to the right of the equilibrium point.

A fourth aspect of the present invention comprises a method for adiabatic quantum computing using a quantum system comprising a plurality of superconducting qubits. The quantum system is capable of being in any one of at least two quantum configurations at any give time. The at least two quantum configurations include a first configuration described by an initialization Hamiltonian H _Oand a second configuration described by a problem Hamiltonian H _P. The Hamiltonian H _Phas a ground state. The method comprises initializing the quantum system to the first configuration. Then the quantum system is adiabatically changed until it is described by the ground state of the problem Hamiltonian H _P. Then the state of the quantum system is read out.

In some embodiments in accordance with the fourth aspect of the invention, each respective first superconducting qubit in the plurality of superconducting qubits is arranged with respect to a respective second superconducting qubit in the plurality of superconducting qubits such that the first respective superconducting qubit and the corresponding second respective superconducting qubit define a predetermined coupling strength. The predetermined coupling strength between each of the first respective superconducting qubits and corresponding second respective superconducting qubits in the plurality of superconducting qubits collectively define a computational problem to be solved. In some instances, the problem Hamiltonian H _Pcomprises a tunneling term for each of the respective superconducting qubits in the plurality of superconducting qubits. The energy of the tunneling term for each respective superconducting qubit in the plurality of superconducting qubits is less than the average of the predetermined coupling strengths between each of the first respective superconducting qubits and second respective superconducting qubits in the plurality of superconducting qubits.

In some embodiments in accordance with the fourth aspect of the invention, the reading out step comprises probing an observable of the σ _XPauli matrix operator or σ _ZPauli matrix operator. In some embodiments, a tank circuit is in inductive communication with all or a portion of the superconducting qubits in the plurality of superconducting qubits. In such embodiments, the reading out step comprises measuring a voltage across the tank circuit. In some embodiments, the superconducting qubit in the plurality of superconducting qubits is a persistent current qubit.

A fifth aspect of the present invention provides a structure for adiabatic quantum computing comprising a plurality of superconducting qubits. The plurality of superconducting qubits is capable of being in any one of at least two configurations at any give time. The at least two configurations include a first configuration characterized by an initialization Hamiltonian H ₀and a second Hamiltonian characterized by a problem Hamiltonian H _P. The problem Hamiltonian has a ground state. Each respective first superconducting qubit in the plurality of superconducting qubits is coupled with a respective second superconducting qubit in the plurality of superconducting qubits such that the first respective superconducting qubit and the corresponding second respective superconducting qubit define a predetermined coupling strength. The predetermined coupling strength between each of the first respective superconducting qubits and the corresponding second respective superconducting qubits collectively define a computational problem to be solved. A tank circuit is inductively coupled to all or a portion of the plurality of superconducting qubits.

In some embodiments in accordance with the fifth aspect of the invention, a superconducting qubit in the plurality of superconducting qubits is a persistent current qubit. In some embodiments, the tank circuit has a quality factor that is greater than 1000. In some embodiments, the tank circuit comprises an inductive element. The inductive element comprises a pancake coil of superconducting material. In some embodiments, the pancake coil of a superconducting material comprising a first turn and a second turn. The superconducting material of the pancake coil is niobium. Furthermore, there is a spacing of 1 about micrometer between the first turn and the second turn of the pancake coil.

In some embodiments in accordance with the fifth aspect of the invention, the tank circuit comprises an inductive element and a capacitive element that are arranged in parallel or in series with respect to each other. In some embodiments, the tank circuit comprises an inductive element and a capacitive element that are arranged in parallel with respect to each other and the tank circuit has an inductance between about 50 nanohenries and about 250 nanohenries. In some embodiments, the tank circuit comprises an inductive element and a capacitive element that are arranged in parallel with respect to each other and the tank circuit has a capacitance between about 50 picofarads and about 2000 picofarads. In some embodiments, the tank circuit comprises an inductive element and a capacitive element that are arranged in parallel with respect to each other and the tank circuit has a resonance frequency between about 10 megahertz and about 20 megahertz. In some embodiments, the tank circuit has a resonance frequency f _Tthat is determined by the equality:
f _T=ω _T/2π=1/√ L _T C _T
such that

- L _Tis an inductance of the tank circuit; and
- C _Tis a capacitance of the tank circuit.

In some embodiments in accordance with the fifth aspect of the invention, the tank circuit comprises one or more Josephson junctions. In some embodiments, the structure further comprises means for biasing the one or more Josephson junctions of the tank circuit. In some embodiments, the structure further comprises an amplifier connected across the tank circuit in such a manner that the amplifier can detect a change in voltage across the tank circuit. In some embodiments, the amplifier comprises a high electron mobility field-effect transistor (HEMT) or a pseudomorphic high electron mobility field-effect transistor (PHEMT). In some embodiments, the amplifier comprises a multi-stage amplifier. In some embodiments, the multi-stage amplifier comprises two, three, or four transistors. In some embodiments, structure further comprises a helium-3 pot of a dilution refrigerator that is thermally coupled to all or a portion of the plurality of superconducting qubits. to.

In some embodiments in accordance with the fifth aspect of the invention, the structure further comprising means for driving the tank circuit by a direct bias current I _DC. In some embodiments, the structure further comprises means for driving the tank circuit by an alternating current I _RFof a frequency ω close to the resonance frequency ω ₀of the tank circuit. In some embodiments, the total externally applied magnetic flux to a superconducting qubit in the plurality of superconducting qubits, Φ _E, is
Φ _E=Φ _DC+Φ _RF

where,

- Φ _RFis an amount of applied magnetic flux contributed to the superconducting qubit by the alternating current I _RF; and
- Φ _DCis an amount of applied magnetic flux that is determined by the direct bias current I _DC. In some embodiments, the structure further comprises means for applying a magnetic field on the superconducting qubit, and wherein
  Φ _DC=Φ _A +f ( t )Φ ₀,

where,

- Φ ₀is one flux quantum;
- f(t)Φ ₀is constant or is slowly varying and is generated by the direct bias current I _DC; and
  Φ _A =B _A ×L _Q,

such that

- B _Ais a magnitude of the magnetic field applied on the superconducting qubit by the means for applying the magnetic field; and
- L _Qis an inductance of the superconducting qubit.
  In some embodiments f(t) has a value between 0 and. In some embodiments, the means for applying a magnetic field on the superconducting qubit comprises a bias line that is magnetically coupled to the superconducting qubit. In some embodiments, the means for applying a magnetic field on the superconducting qubit is an excitation device. In some embodiments, Φ _RFhas a magnitude between about 10 ⁻⁵Φ ₀and about 10 ⁻¹Φ ₀. In some embodiments, the structure further comprises means for varying f(t), Φ _A, and/or Φ _RF. In some embodiments, the structure further comprises means for varying Φ _RFin accordance with a small amplitude fast function. In some embodiments, the means for varying Φ _RFin accordance with a small amplitude fast function is a microwave generator that is in electrical communication with the tank circuit.

In some embodiments in accordance with the fifth aspect of the invention, the structure further comprises an amplifier connected across the tank circuit and means for measuring a total impedance of the tank circuit, expressed through the phase angle χ between driving current I _RFand the tank voltage. In some embodiments, the means for measuring a total impedance of the tank circuit is an oscilloscope.

A sixth aspect of the invention provides a computer program product for use in conjunction with a computer system. The computer program product comprises a computer readable storage medium and a computer program mechanism embedded therein. The computer program mechanism comprises instructions for initializing a quantum system comprising a plurality of superconducting qubits to an initialization Hamiltonian H _O. The quantum system is capable of being in one of at least two configurations at any give time. The at least two configurations include a first configuration characterized by the initialization Hamiltonian H _Oand a second configuration characterized by a problem Hamiltonian H _P. Each respective first superconducting qubit in the plurality of superconducting qubits is arranged with respect to a respective second superconducting qubit in the plurality of superconducting qubits such that the first respective superconducting qubit and the second respective superconducting qubit define a predetermined coupling strength. The predetermined coupling strengths between each of the first respective superconducting qubits and the second respective superconducting qubits collectively define a computational problem to be solved. The computer program mechanism further comprises instructions for adiabatically changing the quantum system until it is described by the ground state of the problem Hamiltonian H _Pand instructions for reading out the state of the quantum system.

In some embodiments in accordance with this sixth aspect of the invention, the computer program mechanism further comprises instructions for repeating the instructions for biasing, instructions for adding, and instructions for adiabatically varying between 1 time and 100 times inclusive. The presence or absence of the voltage response of the tank circuit is observed as an average of the voltage response of the tank circuit to each instance of the instructions for adiabatically changing that are executed by the instructions for repeating.

A seventh aspect of the invention comprises a computer program product for use in conjunction with a computer system. The computer program product comprises a computer readable storage medium and a computer program mechanism embedded therein. The computer program mechanism determines a quantum state of a first target superconducting qubit in a plurality of superconducting qubits. The computer program mechanism comprises instructions for initializing a plurality of superconducting qubits so that they are described by a problem Hamiltonian. The problem Hamiltonian describes (i) the quantum state of the plurality of superconducting qubits and (ii) each coupling energy between qubits in the plurality of qubits. The problem Hamiltonian is at or near a ground state. The computer program mechanism further comprises instructions for adding an rf-flux to the first target superconducting qubit. The rf-flux has an amplitude that is less than one flux quantum. The computer program mechanism further comprises instructions for adiabatically varying an amount of an additional flux in the first target superconducting qubit and observing a presence or an absence of a dip in a voltage response of a tank circuit that is inductively coupled with the first target superconducting qubit during the adiabatically varying step.

In some embodiments in accordance with this seventh aspect of the invention, each superconducting qubit in the plurality of superconducting qubits is in a quantum ground state during all or a portion of the instructions for initializing, instructions for adding, and the instructions for adiabatically varying. In some embodiments, the problem Hamiltonian corresponds to a terminus of an adiabatic evolution of the plurality of superconducting qubits. In some embodiments, the computer program product further comprises instructions for biasing all or a portion of the superconducting qubits in the plurality of superconducting qubits. In such embodiments, the problem Hamiltonian additionally describes the biasing on the qubits in the plurality of superconducting qubits. In some embodiments, an energy of the biasing exceeds the tunneling energy of a tunneling element of the Hamiltonian of a superconducting qubit in the plurality of superconducting qubits thereby causing tunneling to be suppressed in the superconducting qubit during an instance of the instructions for biasing, instructions for adding and the instructions for adiabatically varying.

In some embodiments in accordance with the seventh aspect of the invention, the computer program mechanism further comprises instructions for adiabatically removing additional flux that was added to the first target superconducting qubit during the instructions for adiabatically varying. In some embodiments, the instructions for adiabatically varying comprise instructions for adiabatically varying the additional flux in accordance with a waveform selected from the group consisting of periodic, sinusoidal, triangular, and trapezoidal. In some embodiments, the instructions for adiabatically varying comprise instructions for adiabatically varying the additional flux in accordance with a low harmonic Fourier approximation of a waveform selected from the group consisting of periodic, sinusoidal, triangular, and trapezoidal. In some embodiments, the additional flux has a direction that is deemed positive or negative. In some embodiments, the instructions for adiabatically varying are characterized by a waveform that has an amplitude that grows with time and such that the amplitude of the waveform corresponds to an amount of additional flux that is added to the first target superconducting qubit during an instance of the instructions for adiabatically varying.

In some embodiments in accordance with the seventh aspect of the invention, the additional flux has an equilibrium point that varies with time. In some embodiments, the additional flux is either unidirectional or bidirectional. In some embodiments, the additional flux has a frequency of oscillation between about 1 cycle per second and about 100 kilocycles per second. In some embodiments, the instructions for adding comprise instructions for adding the rf-flux using (i) an excitation device that is inductively coupled to the first target superconducting qubit or (ii) the tank circuit. In some embodiments, the computer program mechanism further comprises instructions for repeating the instructions for adding and the instructions for adiabatically varying between 1 time and 100 times. In such embodiments, the presence or absence of the voltage response of the tank circuit is observed as an average of the voltage response of the tank circuit across each instance of the instructions for adiabatically varying that is executed by the instructions for repeating.

An eight aspect of the invention comprises a computer system for determining a quantum state of a first target superconducting qubit in a plurality of superconducting qubits. The computer system comprises a central processing unit and a memory, coupled to the central processing unit. The memory stores instructions for biasing all or a portion of the qubits in the plurality of superconducting qubits other than the first target superconducting qubit. A problem Hamiltonian describes (i) the biasing on the qubits in the plurality of superconducting qubits and (ii) each coupling energy between respective superconducting qubit pairs in the plurality of superconducting qubits. The problem Hamiltonian is at or near a ground state. The memory further stores instructions for adding an rf-flux to the first target superconducting qubit. The rf-flux has an amplitude that is less than one flux quantum. The memory further stores instructions for adiabatically varying an amount of an additional flux in the first target superconducting qubit and observing a presence or an absence of a dip in a voltage response of a tank circuit that is inductively coupled with the first target superconducting qubit during a time when the instructions for adiabatically varying are executed.

A ninth aspect of the present invention provides a computation device for adiabatic quantum computing comprising a plurality of superconducting qubits. Each superconducting qubit in the plurality of superconducting qubits comprises two basis states associated with the eigenstates of a σ ^ZPauli matrix that can be biased. The quantum computation device further comprises a plurality of couplings. Each coupling in the plurality of couplings is disposed between a superconducting qubit pair in the plurality of superconducting qubits. Each term Hamiltonian for a coupling in the plurality of couplings has a principal component proportional to σ ^Z{circle around (x)}σ ^Z. The sign for at least one principal component proportional to σ ^Z{circle around (x)}σ ^Zfor a coupling in the plurality of couplings is antiferromagnetic. The superconducting qubits and the plurality of couplings are collectively capable of being in any one of at least two configurations. The at least two configurations include a first configuration characterized by an initialization Hamiltonian H ₀and a second Hamiltonian characterized by a problem Hamiltonian H _P. The problem Hamiltonian has a ground state. Each respective first superconducting qubit in the plurality of superconducting qubits is coupled with a respective second superconducting qubit in the plurality of superconducting qubits such that the first respective superconducting qubit and the corresponding second respective superconducting qubit define a predetermined coupling strength. The predetermined coupling strength between each of the first respective superconducting qubits and the corresponding second respective superconducting qubits collectively define a computational problem to be solved. The computation device further comprises a read out circuit coupled to at least one superconducting qubit in the plurality of superconducting qubits.

A tenth aspect of the invention comprises an apparatus comprising a plurality of superconducting charge qubits. Each respective first superconducting charge qubit in the plurality of superconducting charge qubits is coupled with a respective second superconducting charge qubit in the plurality of superconducting charge qubits such that the first respective superconducting charge qubit and the second respective superconducting charge qubit define a predetermined coupling strength. The predetermined coupling strength between each of the first respective superconducting charge qubits and each of the second respective superconducting charge qubits in the plurality of superconducting charge qubits collectively define a computational problem to be solved. Each superconducting charge qubit in the plurality of superconducting charge qubits is capable of being in one of at least two configurations. These at least two configurations include a first configuration in accordance with an initialization Hamiltonian H ₀and a second configuration in accordance with a problem Hamiltonian H _P. The apparatus further comprises an electrometer coupled to a superconducting charge qubit in the plurality of superconducting charge qubits.

In some embodiments in accordance with this tenth aspect of the invention, a superconducting charge qubit in the plurality of superconducting charge qubits comprises (i) a mesoscopic island made of superconducting material, (ii) superconducting reservoir, and (iii) a Josephson junction connecting the mesoscopic island to the superconducting reservoir. In some embodiments, the Josephson junction is a split Josephson junction. In some embodiments, the superconducting charge qubit further comprises a flux source configured to apply flux to the split Josephson junction.

In some embodiments in accordance with the tenth aspect of the invention, the apparatus further comprises a generator capacitively coupled to a superconducting charge qubit in the plurality of superconducting charge qubits by a capacitor. In some embodiments, the generator is configured to apply a plurality of electrostatic pulses to the superconducting charge qubit. The plurality of electrostatic pulses additionally define the computational problem.

In some embodiments in accordance with the tenth aspect of the invention, the apparatus further comprises a variable electrostatic transformer disposed between a first superconducting charge qubit and a second superconducting charge qubit in the plurality of superconducting charge qubits such that the predetermined coupling strength between the first superconducting charge qubit and the second superconducting charge qubit is tunable. In some embodiments, each respective first superconducting charge qubit in the plurality of superconducting charge qubits is arranged with respect to a respective second superconducting charge qubit in the plurality of superconducting charge qubits such that the plurality of superconducting charge qubits collectively form a non-planar graph.

An eleventh aspect of the invention provides a method for computing using a quantum system comprising a plurality of superconducting charge qubits. The quantum system is coupled to an electrometer and the quantum system is capable of being in any one of at least two configurations. The at least two configurations includes a first configuration characterized by an initialization Hamiltonian H ₀and a second configuration characterized by a problem Hamiltonian H _P. The problem Hamiltonian has a ground state. The plurality of superconducting charge qubits are arranged with respect to one another, with a predetermined number of couplings between respective pairs of superconducting charge qubits in the plurality of charge qubits, such that the plurality of superconducting charge qubits, coupled by the predetermined number of couplings, collectively define a computational problem to be solved. The method comprises initializing the quantum system to the initialization Hamiltonian H _O. Then the quantum system is adiabatically changed until it is described by the ground state of the problem Hamiltonian H _P. Next the quantum state of each superconducting charge qubit in the quantum system is read out through the electrometer, thereby solving the computational problem to be solved.

In some embodiments in accordance with the eleventh aspect of the invention, a first superconducting charge qubit in the plurality of superconducting charge qubits is coupled to a second superconducting charge qubit in the plurality of superconducting charge qubits by a capacitor such that the predetermined coupling strength between the first superconducting charge qubit and the second superconducting charge qubit is predetermined and is a function of the physical properties of the capacitor.

In some embodiments in accordance with the eleventh aspect of the invention, a first superconducting charge qubit in the plurality of superconducting charge qubits is coupled to a generator by a device configured to provide a tunable effective charging energy. The device comprises a capacitor and the method further comprises: tuning the value of the effective charging energy of the first superconducting charge qubit by varying the gate voltage on the capacitor of said device. In some embodiments, a superconducting charge qubit in the plurality of superconducting charge qubits comprises a split Josephson junction having a variable effective Josephson energy. In such embodiments, the method further comprises tuning the value of the effective Josephson energy of the superconducting charge qubit by varying a flux applied to the split Josephson junction. In some embodiments, the first configuration is reached by setting the effective Josephson energy of the superconducting charge qubit to a maximum value.

In some embodiments in accordance with the eleventh aspect of the invention, the adiabatically changing step comprises changing the configuration of the system from the first configuration characterized by the initialization Hamiltonian H ₀, to the second Hamiltonian characterized by a problem Hamiltonian H _Pin the presence of tunneling on a superconducting charge qubit in the plurality of superconducting charge qubits.

In some embodiments in accordance with a eleventh aspect of the invention, a first superconducting charge qubit in the plurality of superconducting charge qubits is characterized by (i) an effective Josephson energy that is tunable and (ii) an effective charging energy that is tunable. A minimum value of the effective Josephson energy is less than the effective charging energy of the first superconducting charge qubit A minimum value of the effective Josephson energy is less than a strength of a coupling between the first superconducting charge qubit and a second superconducting charge qubit in the plurality of superconducting charge qubits. The effective charging energy is, at most, equal to a maximum value of the effective Josephson energy of the first superconducting charge qubit. Furthermore, a strength of a coupling between the first superconducting charge qubit and a second superconducting charge qubit in the plurality of superconducting charge qubits is, at most, equal to a maximum value of the effective Josephson energy of the first superconducting charge qubit.

In still another embodiment in accordance with the eleventh aspect of the invention, a first superconducting charge qubit in the plurality of superconducting charge qubits is characterized by (i) an effective Josephson energy that is tunable and (ii) an effective charging energy that is tunable. In such embodiments, the adiabatically changing step comprises adiabatically tuning the effective Josephson energy of the first superconducting charge qubit such that the effective Josephson energy of the first superconducting charge qubit reaches a minimum value when the quantum system is described by the ground state of the problem Hamiltonian H _P.

In some embodiments in accordance with the eleventh aspect of the invention, a first superconducting charge qubit in the plurality of superconducting charge qubits has a first basis state and a second basis state and, when the quantum system is described by the ground state of the problem Hamiltonian H _P, tunneling between the first basis state and the second basis state of the first superconducting charge qubit does not occur.

In some embodiments in accordance with the eleventh aspect of the invention, a first superconducting charge qubit in the plurality of superconducting charge qubits has a first basis state and a second basis state and, when the quantum system is described by the ground state of the problem Hamiltonian H _P, the tunneling between the first basis state and the second basis state of the first superconducting charge qubit does occur. Furthermore, the reading out step comprises probing an observable of the sigma-x Pauli matrix σ ^X.

In some embodiments in accordance with the eleventh aspect of the invention, a first superconducting charge qubit in the plurality of superconducting charge qubits is characterized by (i) an effective Josephson energy that is tunable and (ii) an effective charging energy that is tunable. In such embodiments, a minimum value of the effective Josephson energy is less than the effective charging energy of the first superconducting charge qubit; a minimum value of effective Josephson energy is less than a strength of a coupling between the first superconducting charge qubit and a second superconducting charge qubit in the plurality of superconducting charge qubits; the effective charging energy is greater than a maximum value of the effective Josephson energy of the first superconducting charge qubit; and a strength of a coupling between the first superconducting charge qubit and a second superconducting charge qubit in the plurality of superconducting charge qubits is, at most, equal to the maximum effective Josephson energy of the first superconducting charge qubit. In some such embodiments, the initializing step comprises setting the effective charging energy of the first superconducting charge qubit to a minimum value. In some such embodiments, the adiabatically changing step comprises adiabatically tuning the effective Josephson energy of the first superconducting charge qubit such that the effective Josephson energy is at a minimum value when the quantum system is described by the ground state of the problem Hamiltonian H _P, and adiabatically increasing the effective charging energy of the first superconducting charge qubit.

In some embodiments in accordance with the eleventh aspect of the invention, a first superconducting charge qubit in the plurality of superconducting charge qubits is characterized by an effective Josephson energy that is tunable. The initializing step comprises setting the effective Josephson energy of the first superconducting charge qubit to a minimum value, and the adiabatically changing step comprises (i) adiabatically tuning the effective Josephson energy of the first superconducting charge qubit such that the effective Josephson energy is greater than a minimum value for a period of time before the quantum system is described by the ground state of the problem Hamiltonian H _P, and adiabatically tuning the effective Josephson energy of the first superconducting charge qubit such that the effective Josephson energy is at a minimum value when the quantum system is described by the ground state of the problem Hamiltonian H _P.

4. BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A illustrates a known superconducting qubit.

FIG. 1B illustrates a known energy potential for a superconducting qubit.

FIG. 2 illustrates an exemplary quantum computation circuit model in accordance with the prior art.

FIG. 3 illustrates a known general equation that describes the theory of adiabatic quantum computing.

FIG. 4 illustrates a work flow diagram for a process of adiabatic quantum computing.

FIGS. 5A-5G illustrates arrangements of superconducting qubits for adiabatic quantum computing in accordance with some embodiments of the present invention.

FIGS. 6A-6B illustrates an example of a computational problem that can be solved by adiabatic quantum computing.

FIG. 7A illustrates an energy level diagram for a system comprising a plurality of superconducting qubits during an instantaneous adiabatic change of the system.

FIG. 7B illustrates an energy level diagram for a system comprising a plurality of superconducting qubits when the plurality of qubits are described by the ground state of a problem Hamiltonian H _P.

FIG. 8 illustrates a device for controlling and reading out the state of a superconducting qubit for adiabatic quantum computing in accordance with some embodiments of the present invention.

FIG. 9A illustrates an energy level diagram for a physical system in which the energy level diagram exhibits an anticrossing between energy levels of the physical system in accordance with an embodiment of the present invention.

FIG. 9B illustrates an energy level diagram for a physical system having an energy level crossing in accordance with an embodiment of the present invention.

FIGS. 10A-10D illustrate the waveforms for additional flux Φ _Ain a qubit undergoing adiabatic quantum change, in arbitrary units, in accordance with various embodiments of the present invention.

FIG. 11A illustrates the form of a readout signal for a superconducting qubit having an anticrossing between two energy levels.

FIG. 11B illustrates the form of a readout signal for a superconducting qubit having that does not have anticrossing between two energy levels.

FIGS. 12A-12D illustrate superconducting charge qubits and read out devices in accordance with some embodiments of the present invention.

FIGS. 13A-13B illustrate coupled superconducting charge qubits in accordance with some embodiments of the present invention.

FIGS. 14A-14D illustrates how superconducting qubits can be arranged in accordance some embodiments of the present invention.

FIG. 15 illustrates a system that is operated in accordance with some embodiments of the present invention.

Like reference numerals refer to corresponding parts throughout the several views of the drawings.

5. DETAILED DESCRIPTION OF THE INVENTION

The present invention comprises systems and methods for adiabatic quantum computing using superconducting qubits. In various embodiments of the present invention, adiabatic quantum computing is performed on registers of superconducting qubits that have demonstrated quantum computing functionality. Adiabatic quantum computing is a model of quantum computing that can be used to attempt to find solutions for computationally difficult problems.

General Embodiments

When choosing a candidate system for adiabatic quantum computing there are a few criteria that can be observed. These criteria can be drawn from those described herein below. However, some embodiments of the present invention may not adhere to all of these criteria. One criterion is that the readout device should a Stern-Gerlach σ ^Ztype observation. A second criterion is that the tunneling term in the problem Hamiltonian should be about zero. For H _P=Δσ ^X+εσ ^Zthen Δ≈0. A third criterion is that the magnitude of the tunneling term in the problem, initial, or extra Hamiltonian (H _P, H ₀, H _E) should be tunable. A fourth criterion is that the qubit-qubit coupling should be diagonal in the basis of final qubit states, i.e., σ ^Z{circle around (x)}σ ^Z. Because an Ising model with ferromagnetic couplings has a trivial ground state, all spins aligned, a fifth criterion is that the system have some antiferromagnetic coupling between qubits. Some AFM couplings include the case where all are antiferromagnetic. Also, ferromagnetic couplings have a negative sign −Jσ ^Z{circle around (x)}σ ^Z, and antiferromagnetic couplings have a positive sign Jσ ^Z{circle around (x)}σ ^Z.

Some embodiments of the present invention adhere to the above criteria. Other embodiments of the present invention do not. For instance, in the case of the phase qubit, it is possible to have the tunneling term in the problem Hamiltonian be, not zero, but weak, e.g., for H _P=Δσ ^X+εσ ^Zthen Δ<<ε. In such a case it is possible for the readout device to probe a Stern-Gerlach σ ^Xtype observable. Other embodiments of superconducting adiabatic quantum computers of the present invention do not adhere to the third criterion described above. For example, the magnitude of the tunneling term in the problem, initial, or extra Hamiltonian (H _P, H ₀, H _E) is fixed but the contribution of the problem, initial, or extra Hamiltonian to the instant Hamiltonian is tunable in such embodiments. Specific embodiments of the present invention are described below.

5.1 Exemplary General Procedure

In accordance with embodiments of the present invention, the general procedure of adiabatic quantum computing is shown in FIG. 4. In step 401 , a quantum system that will be used to solve a computation is selected and/or constructed. In some embodiments, each problem or class of problems to be solved requires a custom quantum system designed specifically to solve the problem. Once a quantum system has been chosen, an initial state and a final state of the quantum system need to be defined. The initial state is characterized by the initial Hamiltonian H ₀and the final state is characterized by the final Hamiltonian H _Pthat encodes the computational problem to be solved. In preferred embodiments, the quantum system is initiated to the ground state of the initial Hamiltonian H ₀and, when the system reaches the final state, it is in the ground state of the final Hamiltonian H _P. More details on how systems are selected and designed to solve a computational problem are described below.

In step 403 , the quantum system is initialized to the ground state of the time-independent Hamiltonian, H ₀, which initially describes the quantum system. It is assumed that the ground state of H ₀is a state to which the system can be reliably and reproducibly set. As will be disclosed in further detail below, this assumption is reasonable for quantum systems comprising, at a minimum, specific types of qubits and specific types of arrangements of such qubits.

In transition 404 between steps 403 and 405 , the quantum system is acted upon in an adiabatic manner in order to alter the system. The system changes from being described by Hamiltonian H ₀to a description under H _P. This change is adiabatic, as defined above, and occurs in a period of time T. In other words, the operator of an adiabatic quantum computer causes the system, and Hamiltonian H describing the system, to change from H ₀to a final form H _Pin time T. The change is an interpolation between H ₀and H _P. The change can be a linear interpolation:
H ( t )=(1−γ( t )) H ₀+γ( t ) H _P
where the adiabatic evolution parameter, γ(t), is a continuous function with γ(t=0)=0, and γ(t=T)=1. The change can be a linear interpolation, γ(t)=t/T such that

$H (\frac{t}{T}) = (1 - \frac{t}{T}) H_{0} + \frac{t}{T} H_{P} .$
In accordance with the adiabatic theorem of quantum mechanics, a system will remain in the ground state of H at every instance the system is changed and after the change is complete, provided the change is adiabatic. In some embodiments of the present invention, the quantum system starts in an initial state H ₀that does not permit quantum tunneling, is perturbed in an adiabatic manner to an intermediate state that permits quantum tunneling, and then is perturbed in an adiabatic manner to the final state described above.

In step 405 , the quantum system has been altered to one that is described by the final Hamiltonian. The final Hamiltonian H _Pcan encode the constraints of a computational problem such that the ground state of H _Pcorresponds to a solution to this problem. Hence, the final Hamiltonian is also called the problem Hamiltonian H _P. If the system is not in the ground state of H _P, the state is an approximate solution to the computational problem. Approximate solutions to many computational problems are useful and such embodiments are fully within the scope of the present invention.

In step 407 , the system described by the final Hamiltonian H _Pis read out. The read out can be in the σ ^Zbasis of the qubits. If the read out basis commutes with the terms of the problem Hamiltonian H _p, then performing a read out operation does not disturb the ground state of the system. The read out method can take many forms. The object of the read out step is to determine exactly or approximately the ground state of the system. The states of all qubits are represented by the vector {right arrow over (O)}, which gives a concise image of the ground state or approximate ground state of the system. The read out method can compare the energies of various states of the system. More examples are given below, making use of specific qubits for better description.

5.2 Changing a Quantum System Adiabatically

In one embodiment of the present invention, the natural quantum mechanical evolution of the quantum system under the slowly changing Hamiltonian H(t) carries the initial state H ₀of the quantum system into a final state, the ground state of H _P, corresponding to the solution of the problem defined by the quantum system. A measurement of the final state of the quantum system reveals the solution to the computational problem encoded in the problem Hamiltonian. In such embodiments, an aspect that can define the success of the process is how quickly (or slowly) the change between the initial Hamiltonian and problem Hamiltonian occurs. How quickly one can drive the interpolation between H ₀and H _P, while keeping the system in the ground state

6885325	Sub-flux quantum generator	April, 2005	Omelyanchouk et al.
20020177529	Fluxon injection into annular josephson junctions	November, 2002	Ustinov
20030055513	Method for quantum computing	March, 2003	Raussendorf et al.
20030169041	Quantum computing integrated development environment	September, 2003	Coury et al.
20030224944	Characterization and measurement of superconducting structures	December, 2003	Il'ichev et al.
20040173792	Resonant controlled qubit system	September, 2004	Blais et al.
20040238813	Dressed qubits	December, 2004	Lidar et al.
20050062072	Coupled superconducting charge quantum bit device and controlled-not gate using the same	March, 2005	Yamamoto et al.

JP2002150778	May, 2002	QUANTUM BIT ELEMENT, AND METHOD FOR EXTENDING DECOHERENCE TIME OF QUANTUM BIT
WO-02073229	September, 2002
WO-03021527	March, 2003