Quantum computing execution method and quantum computing execution device

Abstract

A method for performing a quantum computation is provided. The method includes providing a quantum system 300 comprising a component 302. The method includes encoding a computational problem 110 into a problem Hamiltonian 150 for the quantum system. The problem Hamiltonian is a simplex Hamiltonian that is a sum of summand problem Hamiltonians 152. The method includes determining a constraint Hamiltonian 250 for the quantum system. The constraint Hamiltonian is a sum of summand constraint Hamiltonians 252. A ground state of the sum Hamiltonian encodes a solution to the computational problem. The sum Hamiltonian includes a sum of the problem Hamiltonian and the constraint Hamiltonian. The method includes determining a first subset S1 of summand constraint Hamiltonians of the constraint Hamiltonian and a second subset S2 of summand constraint Hamiltonians of the constraint Hamiltonian. The method includes performing N (N≧2) operations. Each round includes preparing an initial quantum state. Each round includes evolving the quantum system according to a sequence of unitary operators, including a problem-encoded unitary operator, a constraint-enforcing unitary operator, and a unitary driving operator. Each problem-encoded unitary operator is a unitary time evolution operator of an augend problem Hamiltonian of the problem Hamiltonian, or is a unitary time evolution operator of a summand problem Hamiltonian of the problem Hamiltonian. Each constraint-enforcing unitary operator is a unitary time evolution operator of an augend constraint Hamiltonian taken from a first subset of augend constraint Hamiltonians of the constraint Hamiltonian, or is a unitary time evolution operator of a summand constraint Hamiltonian taken from the first subset. Each unitary driving operator is a unitary operator that commutes with all augend constraint Hamiltonians from a second subset of augend constraint Hamiltonians of the constraint Hamiltonian. Each round includes performing a measurement of one or more components of the quantum system. The method includes outputting a result 590 of the quantum computation.

Description

[Technical Field]

[0001] Embodiments described herein relate to methods and apparatus for performing quantum computing. The methods use quantum systems that include components, such as qubits. The components of the quantum system are acted upon, for example, by a quantum processing system, to process information carried by the components. Some of the components are measured to reveal information contained in the components. A computational problem is solved based on readouts obtained from the measurements. [Background technology]

[0002] A quantum computing device is a computing device that uses quantum mechanical effects to solve computational problems. In a quantum computing device or quantum computer, information is carried by quantum systems, such as quantum bits ("qubits"). This is in contrast to traditional computers that operate with classical bits, i.e., 0s and 1s. During quantum computing, the quantum systems can be developed to process the quantum bits. For example, a group of qubits in a quantum system can be coupled to each other according to specified interactions. The quantum systems can be developed to process the information carried by the quantum systems to perform computations, i.e., to solve computational problems. Quantum computers are often assisted by classical computers, i.e., computers that operate with classical bits. The classical computer can provide instructions to the quantum computer on how the qubits in the system should be processed by the quantum computer.

[0003] Many approaches to quantum computing require the execution of long-range interactions to perform arbitrary quantum computations. Long-range interactions are interactions that couple qubits that are far apart from each other in a quantum system. Such long-range interactions present an obstacle because they are difficult to realize in practice. In some settings, long-range interactions can be replaced by sequences of short-range interactions. However, these approaches suffer from the drawback that sequences of short-range interactions are inherently sequential, i.e., cannot be parallelized, increasing the execution time of quantum computations. In turn, the fact that such sequences cannot be parallelized may undermine the scalability of quantum computers based on such principles.

[0004] Alternatively, some approaches to quantum computing use only short-range interactions, but they have the drawback of not being fully programmable: such quantum computers are limited in the sense that they are tailored to solve certain computational problems, but cannot solve arbitrary computational problems.

[0005] Still other approaches allow quantum computation to be parallelized to some degree, but this comes at the cost of reducing the efficiency of the quantum computation, i.e., they increase the execution time required by the quantum computer to solve the computational problem at hand.

[0006] Therefore, there is a need for improved methods and apparatus for performing quantum computing. Summary of the Invention

[0007] According to one embodiment, a method for performing quantum computation is provided. The method includes providing a quantum system including components. The method includes encoding a computational problem into a problem Hamiltonian for the quantum system. The problem Hamiltonian is a simplex Hamiltonian that is a sum of summand problem Hamiltonians. The method includes determining a constraint Hamiltonian for the quantum system. The constraint Hamiltonian is a summation of summand constraint Hamiltonians. A ground state of the total Hamiltonian encodes a solution to the computational problem. The total Hamiltonian includes a summation of the problem Hamiltonian and the constraint Hamiltonians. The method includes determining a first subset of summand constraint Hamiltonians of the constraint Hamiltonian and a second subset of summand constraint Hamiltonians of the constraint Hamiltonian. The method includes performing N (N≧2) operations, each of which includes providing an initial quantum state. Each of which includes evolving the quantum system according to a sequence of unitary operators. The sequence includes a problem-encoded unitary operator, a constraint-enforcing unitary operator, and a unitary driving operator. Each problem-encoded unitary operator is a unitary time evolution operator of an augend problem Hamiltonian of the problem Hamiltonian, or a unitary time evolution operator of a summation of augend problem Hamiltonians of the problem Hamiltonian. Each constraint-enforcing unitary operator is a unitary time evolution operator of an augend constraint Hamiltonian obtained from a first subset of augend constraint Hamiltonians of the constraint Hamiltonian, or a unitary time evolution operator of a summation of augend constraint Hamiltonians obtained from the first subset. Each unitary driving operator is a unitary operator that commutes with all augend constraint Hamiltonians from a second subset of augend constraint Hamiltonians of the constraint Hamiltonian. Each round includes performing measurements of one or more components of the quantum system. The method includes outputting a result of the quantum computation.

[0008] According to a further embodiment, an apparatus for performing quantum computation is provided. The apparatus includes a quantum system with components. The apparatus includes a classical computing system. The classical computing system is configured to encode a computational problem into a problem Hamiltonian for the quantum system. The problem Hamiltonian is a simplicial Hamiltonian that is a sum of summand problem Hamiltonians. The classical computing system is configured to determine a constraint Hamiltonian for the quantum system, the constraint Hamiltonian being a summation of summand constraint Hamiltonians. A ground state of the total Hamiltonian encodes a solution to the computational problem. The total Hamiltonian includes a summation of the problem Hamiltonian and the constraint Hamiltonians. The classical computing system is configured to determine a first subset of summand constraint Hamiltonians of the constraint Hamiltonian and a second subset of summand constraint Hamiltonians of the constraint Hamiltonian. The apparatus includes a quantum processing system with a unitary evolution device and a measurement device. The quantum processing system is configured to perform N (N≧2) operations, each of which involves evolving the quantum system by a unitary evolution device according to a sequence of unitary operators, the sequence including a problem-encoded unitary operator, a constraint-enforcing unitary operator, and a unitary driving operator. Each problem-encoded unitary operator is either a unitary time evolution operator of an augend problem Hamiltonian of the problem Hamiltonian, or a unitary time evolution operator of a summation of augend problem Hamiltonians of the problem Hamiltonian. Each constraint-enforcing unitary operator is either a unitary time evolution operator of an augend constraint Hamiltonian obtained from a first subset of augend constraint Hamiltonians of the constraint Hamiltonian, or a unitary time evolution operator of a summation of augend constraint Hamiltonians obtained from the first subset. Each unitary driving operator is a unitary operator that commutes with all summand constraint Hamiltonians from a second subset of summand constraint Hamiltonians of the constraint Hamiltonians. Each round includes performing, with a measurement device, a measurement of one or more components of the quantum system. The classical computation is further configured to output a result of the quantum computation.

[0009] Embodiments also relate to methods of operating the systems described herein and to the use of the systems to perform methods according to embodiments described herein.

[0010] Further advantages, features, aspects and details that can be combined with the embodiments described herein are evident from the dependent claims, the description and the drawings. [Brief explanation of the drawings]

[0011] A full and enabling disclosure, to one of ordinary skill in the art, is set forth more particularly in the remainder of the specification, including reference to the accompanying drawings, in which: [Figure 1] We show the encoding of the computational problem into the problem Hamiltonian. [Figure 2] Denote the constraint Hamiltonian, which is a sum of the augend constraint Hamiltonians. The augend constraint Hamiltonians are grouped into a first set of augend constraint Hamiltonians and a second set of augend constraint Hamiltonians. [Figure 3] 1 shows the spatial configuration of a first set of summand-constrained Hamiltonians acting on a quantum system containing components. [Figure 4] We show the decomposition of a quantum system into subsystems based on the spatial arrangement of a first set of summand-constrained Hamiltonians. [Figure 5] 1 illustrates an apparatus for performing quantum computing according to embodiments described herein. [Figure 6] Here is an example of modularizing a parity-compiled computation problem. [Figure 7] Here is an example of a complete parity-coded graph: [Figure 8] We present examples of configurations of summand-constrained Hamiltonians with division into three- and four-body regimes. [Figure 9] We present an example of an optimized set of explicitly enforced constraints so that a hybrid drive line that preserves the remaining constraints can be implemented in a shallow-depth parallelizable quantum circuit. [Figure 10]We demonstrate modularization of the layout of qubits within a grid with additional explicitly enforced constraints. [Figure 11] We show the depth of the quantum circuit for executing a single step of the QAOA protocol for the layout shown in Figure 7. [Figure 12] We show the average residual energy after optimization and the relative amount of explicitly enforced constraint terms for different system sizes. [Figure 13] We present a possible decomposition of the unitary operator e-iβX^(μ) corresponding to the time evolution under the driving term into a CNOT gate and an Rx rotation gate. [Figure 14] 1 shows an example of two sets of connected drive lines in a submodule with assigned priorities. DETAILED DESCRIPTION OF THE INVENTION

[0012] Reference will now be made in detail to various exemplary embodiments, one or more examples of which are illustrated in the figures, each example being provided by way of illustration and not by way of limitation. For example, features illustrated or described as part of one embodiment can be used on or in conjunction with other embodiments to yield still further embodiments. This disclosure is intended to include such modifications and variations.

[0013] In the following description of the drawings, the same reference numbers refer to the same components. Generally, only the differences with respect to individual embodiments will be described. The structures shown in the drawings are not necessarily drawn to scale and may include details that are depicted in an exaggerated manner to allow a better understanding of the embodiments.

[0014] SUMMARY OF THE INVENTION

[0003] Embodiments described herein relate to methods and apparatus for performing gate-based quantum computing. Gate-based quantum computing, or digital quantum computing, can be understood as a computational method in which quantum computation is driven by a sequence of unitary operators. Gate-based quantum computing is distinguished from other approaches such as adiabatic quantum computing (quantum annealing) or measurement-based quantum computing.

[0015] A quantum system as described herein is a physical system that exhibits quantum effects. That is, a quantum system is a real-world object. A quantum system includes components. Components are physical quantum entities themselves, which can be thought of as smaller d-level quantum systems that jointly form the quantum system. Specifically, components of a quantum system can be qubits. A qubit should be understood as a physical entity that realizes a two-level quantum system. Components can be d-level quantum systems ("qudits") with d>2, or only two of the d levels can be used.

[0016] A quantum system can be in different quantum states, such as an initial quantum state (the quantum state prepared at the start of a quantum computation) and a final quantum state (the quantum state that the quantum computation ultimately ends up in). The final quantum state can be or approximate the ground state of a Hamiltonian for the quantum system, such as the total quantum Hamiltonian described herein. A quantum system can evolve from its initial quantum state toward or to the ground state of the total quantum Hamiltonian by executing a sequence of unitary operators. Such evolution is a real-world process, particularly a controlled technological process (quantum computation) that leads a quantum system from an initial quantum state to an a priori unknown final quantum state that contains information about the solution to a computational problem. The information can be revealed by measuring the quantum system or parts of it, i.e., at least some of its components. The act of measurement is also a physical / technical process. Measurement allows one to obtain a readout of a quantum system. A readout of a quantum system is a set of measurements obtained by measuring components of a quantum system, involving physical interactions with the components of the quantum system.

[0017] A quantum system can include K components, which can be qubits, where K can be at least 100, at least 1000, or at least 10,000. K can range from 100 to 10,000, or from 100 to 100,000, although K can be greater than 100,000. It should be understood that the quantum systems shown and illustrated in the figures can be much smaller for purposes of illustration and explanation and are not intended to be limiting in any way.

[0018] If H^ is the Hamiltonian of a quantum system, then the operator exp(itH^) is a unitary operator, where t is the time parameter. A unitary operator of the form exp(itH^) will be referred to herein as a unitary time evolution operator, or unitary time evolution for short, according to the Hamiltonian H^. A quantum system may evolve by the unitary time evolution of a Hamiltonian. The act of performing a unitary operator is a physical / technical process. The evolution of a quantum system by the unitary time evolution exp(itH^) may involve turning on interactions between a subset of the components of the quantum system, where the interactions are defined by the Hamiltonian H^. The interactions may be turned on for a period of time t. The interactions may be turned off after the period t has elapsed.

[0019] In realistic systems, at least a small amount of noise is always present. Therefore, quantum states cannot be realized with 100% accuracy. Similarly, operations performed in quantum systems, such as unitary operators and measurements, are always subject to at least some noise and are not realized with 100% accuracy. It should be understood that quantum states and operations described herein encompass states and operations that are subject to small amounts of noise.

[0020] According to one embodiment, a method for performing quantum computation is provided. The method includes providing a quantum system including components. The method includes encoding a computational problem into a problem Hamiltonian for the quantum system. The problem Hamiltonian is a simplex Hamiltonian that is a sum of summand problem Hamiltonians. The method includes determining a constraint Hamiltonian for the quantum system. The constraint Hamiltonian is a summation of summand constraint Hamiltonians. A ground state of the total Hamiltonian encodes a solution to the computational problem. The total Hamiltonian includes or is a summation of the problem Hamiltonian and the constraint Hamiltonians. The method includes determining a first subset of summand constraint Hamiltonians of the constraint Hamiltonian and a second subset of summand constraint Hamiltonians of the constraint Hamiltonian. The method includes performing N operations, where N≧2, each operation including providing an initial quantum state. Each iteration involves evolving the quantum system according to a sequence of unitary operators, the sequence including or consisting of a problem-encoded unitary operator, a constraint-enforcing unitary operator, and a unitary driving operator. Each problem-encoded unitary operator is a unitary time evolution operator of an augend problem Hamiltonian of the problem Hamiltonian, or is a unitary time evolution operator of a summand problem Hamiltonian of the problem Hamiltonian. Each constraint-enforcing unitary operator is a unitary time evolution operator of an augend constraint Hamiltonian taken from a first subset of augend constraint Hamiltonians of the constraint Hamiltonian, or is a unitary time evolution operator of a summand constraint Hamiltonian taken from the first subset. Each unitary driving operator is a unitary operator that commutes with all augend constraint Hamiltonians from a second subset of augend constraint Hamiltonians of the constraint Hamiltonian. Each iteration includes performing a measurement of one or more components of the quantum system. The method includes outputting a result of the quantum computation.

[0021] According to embodiments described herein, the (a priori unknown) solution to a computational problem is encoded in the basis space of the total Hamiltonian. To determine the solution, the quantum system evolves toward the ground state of the total Hamiltonian by unitary evolution, more specifically, by applying a sequence of unitary operators over N operations. The N operations provide an iterative process in which the quantum system gradually approaches the ground state of the total Hamiltonian. A final measurement of the state of the quantum system at the end of the iterative process reveals the solution to the computational problem.

[0022] The total Hamiltonian can be the sum of two parts: the problem Hamiltonian and the constraint Hamiltonian. Therefore, the ground state of the total Hamiltonian can be characterized as a quantum state with low energy with respect to both the problem Hamiltonian and the constraint Hamiltonian. Therefore, by lowering the energy of the quantum system with respect to both the problem Hamiltonian and the constraint Hamiltonian, the quantum system can evolve toward the ground state of the total Hamiltonian. By applying a problem-encoded unitary operator to a sequence of N unitary operators (the problem-encoded unitary operator is the time evolution of the summand problem Hamiltonian), the quantum system evolves into a region of states with low energy with respect to the problem Hamiltonian. With respect to the constraint Hamiltonian, the summand constraint Hamiltonian is divided into two groups: a first subset (denoted S1) and a second subset (denoted S2) of the summand constraint Hamiltonian. The summand constraint Hamiltonians from the first subset S1 are treated similarly to the summand problem Hamiltonians. That is, by performing a unitary time evolution (these time evolution operators are constraint-enforcing unitary operators) of (the summand problem Hamiltonians) from the first subset S1, the quantum system evolves to a quantum state that has low energy with respect to each of the constraint Hamiltonians from the first subset S1. The summand constraint Hamiltonians in the first subset S1 are said to be "explicitly" enforced. In contrast, the summand constraint Hamiltonians from the second subset S2 are not explicitly enforced. Rather, a set of unitary operators (unitary driving operators) is selected. This is such that when evolving a quantum system according to a unitary driving operator, the energy of the quantum system with respect to the summand-constrained Hamiltonians of the second subset S2 is conserved (i.e., the unitary driving operator commutes with all summand-constrained Hamiltonians from the second subset S2).Thus, if a quantum system begins in the ground state of the summand-constrained Hamiltonian of the second subset S2, the quantum system will remain within the ground state of said summand-constrained Hamiltonian throughout the evolution of the quantum system, and so there is no need to explicitly enforce the summand-constrained Hamiltonian of the second subset S2. The summand-constrained Hamiltonian of the second subset S2 is said to be "implicitly" enforced.

[0023] Thus, the present disclosure provides a "hybrid" approach in which some summand constraint Hamiltonians are explicitly enforced and others are implicitly enforced. Explicit enforcement of the summand constraint Hamiltonians of the first subset S1 has the advantage that the corresponding constraint-enforcing unitary operators are highly parallelizable, i.e., these unitary operators can be implemented with a small circuit depth, significantly facilitating practical implementation. Nevertheless, not all summand constraint Hamiltonians of the constraint Hamiltonians are explicitly enforced. This is because explicit enforcement would increase the size of the subspace of quantum states searched during the iterative process described above, thereby increasing the execution time of the computation. In contrast, implicit enforcement of the second subset S2 of summand constraint Hamiltonians, by its very structure, forces the quantum system to remain within the basis space of the summand constraint Hamiltonians of the second subset S2, thereby limiting the size of the subspace of the quantum system investigated during the quantum computation. Thus, the embodiments described herein provide a combination of two advantages. Namely, a high degree of parallelizability combined with a smaller search space (due to the explicit enforcement of the summand constraint Hamiltonian from the first subset S1) and thus improved execution time of the computation (due to the implicit enforcement of the summand constraint Hamiltonian from the second subset S2).

[0024] calculation problem

[0025] The computational problem may be a decision problem, an optimization problem, or another type of computational problem. The computational problem may be, for example, any one of a variety of computational problems considered in the fields of computer science, physics, chemistry, or engineering. The computational problem may be an NP-hard problem, such as an Ising spin model problem. The computational problem of the present disclosure may be any computational problem such as those described in EP 3113084, which is incorporated herein by reference.

[0026] The size of a computational problem can be understood as a measure of the number of classical information units, e.g., the number of classical bits, required to specify the computational problem. The size of a computational problem can depend on or be the number of input variables of the computational problem. As the number of input variables increases, the size of the computational problem can also increase.

[0027] Problem Hamiltonian

[0028] H^ P The Hamiltonian in question, denoted by , is the simplicial Hamiltonian of the quantum system. A simplicial Hamiltonian is a Hamiltonian in which no interactions occur between groups of two or more components. A simplicial Hamiltonian can represent interactions between components of a quantum system and an external entity, such as a magnetic or electric field. Each component interacts with the external entity individually.

[0029] The problem Hamiltonian is a sum of summand problem Hamiltonians. Each summand problem Hamiltonian can act on a single component of the quantum system. The problem Hamiltonian is H^ P =Σ k H^ P,k where each Ĥ P,k is the summand problem Hamiltonian acting only on the kth component of the quantum system.

[0030] The problem Hamiltonian can have tunable parameters. The tunable parameters of the problem Hamiltonian can be parameters that represent the strength and / or direction of the interaction between the components of the quantum system and an external entity. The external entity can be a field, particularly a simplicial field. A simplicial field may refer to a field that affects a single component of the quantum system. The external entity can include, for example, one or more magnetic fields, one or more electric fields, one or more laser fields, one or more microwaves, one or more phase shifts due to mechanical deformation, or a combination thereof. The tunable parameters of the problem Hamiltonian can include multiple field strengths and / or multiple field directions of the simplicial field acting on the components of the quantum system.

[0031] The problem Hamiltonian is H^ P =Σ k J k σ^ z (k) where σ̂ z (k) is the Pauli operator of the kth component of the quantum system, and each J k is a coefficient. Coefficient J k can form adjustable parameters of the problem Hamiltonian. Each term J k σ^ z (k) may be the summand problem Hamiltonian described herein.

[0032] Figure 1 shows the problem Hamiltonian H^ P =Σ k H^ P,k Here is the computational problem 110 coded as follows: P and each summand problem Hamiltonian H^ P,k are shown in FIG. 1 by reference numerals 150 and 152, respectively.

[0033] Encoding the computational problem into a problem Hamiltonian may include determining, from the computational problem, a problem encoding configuration of adjustable parameters of the problem Hamiltonian. The problem encoding configuration may include information about the computational problem. In particular, there may be a one-to-one correspondence between the computational problem and the problem encoding configuration. For example, a problem of the form Ĥ P =Σ k J k σ^ z (k) In the case of the Hamiltonian problem, the coefficient J k may form tunable parameters, and the problem encoding construct may be a set of parameters J that encode the computational problem to be solved by quantum computing. k It can be a particular set of values.

[0034] Encoding a computational problem into a problem Hamiltonian can involve a two-stage process in which the computational problem is first mapped to an auxiliary computational problem, and then the auxiliary computational problem is mapped to the problem Hamiltonian.

[0035] Encoding the computational problem into the problem Hamiltonian can include mapping the computational problem to an auxiliary computational problem, where the auxiliary computational problem includes determining a ground state of a spin model, such as an Ising spin model. The auxiliary computational problem can be an Ising spin model problem. The auxiliary computational problem can be an NP-hard computational problem, such as an Ising spin model problem. Mappings from various computational problems to Ising spin model problems and other NP-hard problems are known in the literature.

[0036] Encoding the computational problem into the problem Hamiltonian can include determining the problem Hamiltonian from an auxiliary computational problem. Specifically, the problem encoding configuration of the adjustable parameters of the problem Hamiltonian can be determined from the auxiliary computational problem. For example, each interaction between spins in the spin model of the auxiliary computational problem can be mapped to an augend problem Hamiltonian of the problem Hamiltonian. A specific encoding (called "parity" encoding) that enables the problem Hamiltonian to be determined from an Ising spin model problem is described in EP 3113084 and WO 2022 / 008057. WO 2022 / 008057 is incorporated herein by reference.

[0037] Constraint Hamiltonian

[0038] The act of determining the constraint Hamiltonian can include determining a classical description of the constraint Hamiltonian. Determining can include computing (e.g., by a classical computing system), reading (e.g., from a memory), receiving (e.g., over a communication channel), etc. The acts of determining the first and second subsets of the summand constraint Hamiltonian can be understood similarly.

[0039] Constraint Hamiltonian (H^ C ) can be a short-range Hamiltonian. A short-range Hamiltonian may refer to a Hamiltonian that describes the bonding interactions within a group of components, with an interaction cutoff distance D SR No interaction occurs between components that are separated from each other by a distance greater than the interaction cutoff distance D SR may be a constant distance. SRcan be much smaller than the maximum component distance between components in a quantum system. For example, the interaction cutoff distance can be 30% or less, 20% or less, or 10% or less of the maximum component distance. When components are arranged in a lattice with a fundamental distance (lattice constant), the short-range quantum Hamiltonian can be one in which no interaction occurs between components separated by a distance greater than r times the fundamental distance (lattice constant) of the lattice, where r can be between 1 and 5. For example, r can be √2, 2, 3, 4, or 5. The short-range Hamiltonian H^ can be expressed as H^=Σ i H^ i where each Ĥ i is the summand Hamiltonian of H^, and each summand Hamiltonian H^ i is the distance between any two members of a group that is the interaction cutoff distance D SR It acts only within a group of components of a quantum system, such that they are separated from each other by a distance equal to: i is H^ i =K^ i It can have the form [×]I, where [×] is the tensor product and K^ i is an operator that operates within a group of components, and I is an identity operator that operates on all components outside said group of components.

[0040] The constraint Hamiltonian may be a d-body Hamiltonian, where d is 8 or less, especially 4 or less. A d-body Hamiltonian can refer to a Hamiltonian that describes interactions between multiple components, with no bonding interactions occurring between groups containing d+1 or more components. A d-body Hamiltonian is a summation of augend Hamiltonians, each of which describes bonding interactions between groups of d or fewer components.

[0041] The Z-type operators are Σ j a j Z^ j (including the case where the sum contains only one term), where each a j are the coefficients, and each Z^ jis the Pauli σ^ z Tensor product of operators, or a single Pauli σ^ z operator. The constraint Hamiltonian can also be a Z-type operator. The constraint Hamiltonian is H^ C =Σ l C^ l Each C^ can have the form l is C^ l =a l Z^ l +b l It has the form I and Z^ l is Pauli σ^ z is the tensor product of the operators, I is the identity operator, and a l and b l is the coefficient. Each C^ l can be the summand constraint Hamiltonian.

[0042] Specific forms of Hamiltonians (problem Hamiltonian, constraint Hamiltonian, driving Hamiltonian, etc.) are provided herein as examples. For example, as mentioned above, the problem Hamiltonian and the constraint Hamiltonian can be expressed as the Pauli σ̂ z It should be understood that this choice of type of Pauli operator does not lose generality in that the corresponding directions (x, y, z) can be chosen freely or the types of Pauli operators can be permuted. The problem Hamiltonian and the constraint Hamiltonian can use the same type of Pauli operator.

[0043] Constraint Hamiltonian H^ C is the total Hamiltonian (H^ total (denoted by ) has the property that the ground state encodes the solution to the computational problem. Here, the total Hamiltonian is the problem Hamiltonian H^ P and the constraint Hamiltonian H^ C and H^ total =H^ P +H^ CThe ground states of the total Hamiltonian can be understood to encode a solution to a computational problem in the sense that the ground states contain information about the solution to the problem. This information is revealed by performing one or more measurements on the ground states. Based on the results of the measurements, a solution (e.g., a trial solution) to the computational problem can be determined.

[0044] The term "constraint Hamiltonian" derives from the property that encoding an Ising model problem (which may be the original computational problem or an auxiliary computational problem onto which the original computational problem is mapped) into a problem Hamiltonian can increase the number of degrees of freedom. This means that only the basis space of the problem Hamiltonian contains quantum states that do not correspond to spin configurations of the Ising model, i.e., quantum states that cannot be "mapped back" to the Ising model. To eliminate these additional degrees of freedom, a constraint Hamiltonian is introduced. The constraint Hamiltonian imposes an energy penalty or energy constraint on the aforementioned quantum states so that the basis space of the sum of the problem Hamiltonian and the constraint Hamiltonian, i.e., the total Hamiltonian, contains only quantum states that correspond to solutions to the computational problem. Specifically, each summand constraint Hamiltonian can impose a parity constraint on a subgroup of components such that the number of components in the quantum state |1> within that subgroup is even.

[0045] A particular encoding that allows determining the problem Hamiltonian and the corresponding constraint Hamiltonians from an Ising spin model problem is described in EP 3113084 and WO 2022 / 008057.

[0046] As mentioned above, the total Hamiltonian can be the sum of the problem Hamiltonian and the constraint Hamiltonians. In other embodiments, the total Hamiltonian can include additional terms; in other words, the total Hamiltonian can include any additional terms in addition to the sum of the problem Hamiltonian and the constraint Hamiltonians. The additional terms may correspond, for example, to additional conditions ("side conditions") imposed on the solution to the computational problem.

[0047] Figure 2 shows the constraint Hamiltonian H^ C =Σ l C^ l The constraint Hamiltonian H^ C is the summand constraint Hamiltonian C^ l (In the example shown in FIG. 2, there are seven summand constraint Hamiltonians, but this number is for illustrative purposes only and the disclosure is not limited thereto.) The constraint Hamiltonian and each summand constraint Hamiltonian are indicated in FIG. 2 by reference numerals 250 and 252, respectively. The set of summand constraint Hamiltonians 252 is divided into a first subset S1 and a second subset S2. In the illustrated example, the first subset S1 consists of summand constraint Hamiltonians C^1, C^2, and C^3, and the second subset S2 consists of summand constraint Hamiltonians C^4, C^5, C^6, and C^7. Again, this example is for illustrative purposes only and the disclosure is not limited thereto.

[0048] The first and second subsets of summand constraint Hamiltonians of the constraint Hamiltonian are denoted herein as S1 and S2, respectively. The first subset S1 and the second subset S2 may be disjoint subsets. The union of the first subset S1 and the second subset S2 may form the entire set of summand constraint Hamiltonians of the constraint Hamiltonian. The first subset S1 and the second subset S2 may form a partition of the summand constraint Hamiltonians of the constraint Hamiltonian. Specific examples of the first and second subsets of summand constraint Hamiltonians are described in the "Further Aspects" section below.

[0049] N operations

[0050] N operations may include 10 or more, specifically 100 or more, more specifically 1000 or more, or even 100000 or more operations.

[0051] Each round of N operations includes preparing an initial quantum state for that round. The initial quantum state may be the same for all N operations. Alternatively, different initial quantum states may be prepared for different rounds of operations.

[0052] The initial quantum states of at least some, and possibly all, of the N operations may be ground states of a partially constrained Hamiltonian. The partially constrained Hamiltonian is the sum of all summand constrained Hamiltonians obtained from the second subset S2 (the partially constrained Hamiltonian is the "second partially constrained Hamiltonian" described further below). The partially constrained Hamiltonian has a basis space. The basis space consists of all quantum states that are ground states of the partial Hamiltonian. The basis space has a ground space basis, i.e., an orthonormal basis, consisting of a set of quantum basis states. For example, each quantum basis state may have the form |x>, where x is a bit string. That is, each quantum basis state may be a computational basis state (a standard basis state). The initial quantum states of at least some, and possibly all, of the N operations may be a superposition of all quantum basis states of the ground space basis. Specific examples of initial quantum states are described in the "Further Aspects" section below.

[0053] Each of the N operations involves evolving the quantum system according to a sequence of unitary operators, including a problem-encoding unitary operator, a constraint-enforcing unitary operator, and a unitary driving operator.

[0054] Each problem-encoded unitary operator is a unitary time evolution operator of the summand problem Hamiltonian of the problem Hamiltonian, or a unitary time evolution operator of the summation of the summand problem Hamiltonians of the problem Hamiltonian. A problem-encoded unitary operator may have the form exp(itÂ), where t is a coefficient and  is the single summand problem Hamiltonian Ĥ. P,k or two or more summands problem Hamiltonian H^ P,k (A^ is equal to the sum of all the summands in the Hamiltonian H^P,k Since A^ is the sum of H^ P (including when it is equal to ).

[0055] Each constraint forcing unitary operator is a unitary time evolution operator of an augend constraint Hamiltonian taken from a first subset S1 of augend constraint Hamiltonians of the constraint Hamiltonian, or is a unitary time evolution operator of a summation of augend constraint Hamiltonians taken from said first subset S1. C is expressed by the following formula:

number

[0056] Each constrained unitary operator can act trivially in the underlying spatial basis of the second partially constrained Hamiltonian, where an operator is considered to act trivially in an underlying spatial basis if the operator maps each quantum basis state of the underlying spatial basis to itself up to a proportionality factor.

[0057] Each unitary driving operator is a summand-constrained Hamiltonian C^ from the second subset S2 of summand-constrained Hamiltonians. j Each unitary driving operator commutes with one or more summand constraint Hamiltonians C^ from the first subset S1. i Each unitary driving operator may have the form exp(itH^), where t is a coefficient and H^ is Σ j b j X^ j is an operator of the form (including the case where the sum contains only one term). j are the coefficients, and each X^ j is the Pauli σ X Tensor product of operators or a single Pauli σ X is an operator.

[0058] Each unitary driving operator may have a nontrivial action in the base spatial basis of the second partially constrained Hamiltonian. An operator is considered to have a nontrivial action in the base spatial basis if the operator does not act trivially in the base spatial basis. For each quantum basis state |x> in the base spatial basis of the second partially constrained Hamiltonian, each unitary driving operator may map |x> to a linear combination of two or more quantum basis states in the base spatial basis of the second partially constrained Hamiltonian.

[0059] Methods according to embodiments described herein can include determining a driving Hamiltonian that is a sum of augend driving Hamiltonians, where the driving Hamiltonian commutes with all of the augend constraint Hamiltonians of the second subset S2. Each unitary driving operator can be a unitary time evolution operator of an augend driving Hamiltonian of the driving Hamiltonian, or a unitary time evolution operator of a sum of two or more augend driving Hamiltonians of the driving Hamiltonian (including when the unitary driving operator is a unitary time evolution operator of the driving Hamiltonian and the driving Hamiltonian is a sum of all the augend driving Hamiltonians).

[0060] The driving Hamiltonian can be an X-type operator. The X-type operator is Σj b j X^ j (including the case where the sum contains only one term), where each b j are the coefficients, and each X^ j is Pauli σ^ x Tensor product of operators or a single Pauli σ^ x operator. It should again be understood that this choice of Pauli operator is without loss of generality in that the corresponding directions (x, y, z) can be chosen freely or the type of Pauli operator can be permuted. The driving Hamiltonian may use a different type of Pauli operator than the problem Hamiltonian and the constraint Hamiltonian. In the "Further Aspects" section below, the driving Hamiltonian will be referred to as a "hybrid driving Hamiltonian."

[0061] The sequence of unitary operators for a set of operations is U^1, U^2, ···, U^ m where each U^ i is a unitary operator. U^ i At least some, possibly all, of U^1, U^2, , U^2 may be problem-encoding unitary operators, constraint-enforcing unitary operators, or unitary-driven operators. The initial state of a problem turn can be represented by |ψ>. m Evolving a quantum system according to the sequence implies that the quantum state of the quantum system after the sequence is applied is, at least approximately, U^ m This can be understood to mean that U^2U^1|ψ>.

[0062] If a quantum system has the sequence U^1, U^2, . . . , U^ m The fact that the sequence evolves according to i This does not mean that any operator U^ is implemented as a single unitary operator. i can itself be implemented as a product of multiple unitary operators (quantum gates) or as a circuit. This is because the unitary operator U^ iThis can be advantageous when is too complex to be implemented as a single unitary operator. i as a quantum circuit of some simpler unitary operators (e.g., short-distance d-field unitary operators with a small constant d), we obtain the unitary operation U^ i may be easier to implement.

[0063] Evolving a quantum system according to a sequence of unitary operators may include executing at least some of the unitary operators of the sequence by a quantum circuit comprising a plurality of quantum gates. At least some, in particular all, of the problem-encoded unitary operators, constraint-enforcing unitary operators and unitary-driving operators of the sequence may be executed by the quantum circuit. The term "quantum circuit" refers to a logic gate circuit comprising logic gates, each logic gate being a unitary operator (referred to in this context as a "quantum gate").

[0064] Each quantum gate of a quantum circuit may be a short-distance unitary operator. A short-distance unitary operator is a unitary operator that acts only within a subgroup of components of a quantum system, where any two components within the subgroup are separated by a distance less than or equal to the interaction cutoff distance of the quantum system, as described herein. A short-distance unitary operator does not act on components outside of said subgroup of components.

[0065] Additionally or alternatively, each quantum gate of a quantum circuit may be a d-field unitary operator, where d may be a small constant. For example, d may be equal to or less than 8, or even equal to or less than 4. A d-field unitary operator refers to a unitary operator that operates only within a subgroup containing at most d components of a quantum system. A d-field unitary operator does not operate on components outside said subgroup. A simplicial unitary operator is a d-field unitary operator with d=1.

[0066] The sequence of unitary operators for at least some of the N operations may include K problem-encoding unitary operators, and / or L constraint-enforcing unitary operators, and / or M unitary-driven operators, where K, L, and / or M may be 5 or greater, particularly 10 or greater, and even more particularly 200 or greater.

[0067] For each of the N operations, evolving the quantum system according to a sequence of unitary operators in that operation may include applying the sequence of unitary operations to an initial quantum state in that operation.

[0068] The sequence of unitary operators for at least some of the N operations is A1A2···A p or can include at least a subsequence of the form, where p≧3, and specifically p is 10 or more, 100 or more, or 1000 or more. i is X i Y i Z i is a product of the form X i , Y i and Z i One of the is a problem-encoding unitary operator, X i , Y i and Z i Another one of is a constraint-enforcing unitary operator, X i , Y i and Z i Yet another one of is a unitary driving operator. Examples of possible sequences of unitary operators are discussed in more detail in the "Further Aspects" section below.

[0069] Each of the N operations includes performing a measurement of one or more components of the quantum system. For each of the N operations, the measurement may be performed on a quantum state resulting from evolving the quantum system according to the sequence of unitary operators in the operation. Performing the measurement of one or more components may include evolving a Pauli operator, e.g., a Pauli operator σ, for each of the one or more components. zThis may include measuring

[0070] In some embodiments, the method includes information feedforward, where the sequence of unitary operators applied in the series of operations may depend on a measurement result of a measurement performed in one or more previous operations, e.g., at least two previous operations. The N operations may include one or more adaptive operations, e.g., 10 or more, 100 or more, 1000 or more, or even 100,000 or more adaptive operations. For each adaptive operation, a unitary operator in the sequence of unitary operators in the adaptive operation may be determined based on at least one measurement result of a measurement performed in a previous one of the N operations.

[0071] The N operations can include a first operation. Evolving a quantum system according to the sequence of unitary operators in the first operation can result in a first quantum state of the quantum system. Performing a measurement in the first operation can include measuring the energy of the first quantum state. Measuring the energy of a quantum state, such as the first quantum state, can include measuring a Hamiltonian, such as a total Hamiltonian described herein.

[0072] The N operations can include a second operation performed after the first operation. Evolving the quantum system according to the sequence of unitary operators in the second operation can result in a second quantum state of the quantum system. Performing a measurement in the second time can include measuring the energy of the second quantum state. Measuring the energy of the second quantum state can include measuring a Hamiltonian, such as a total Hamiltonian.

[0073] The methods described herein may include comparing an energy of a first quantum state to an energy of a second quantum state. The methods may also include determining a sequence of unitary operators to be applied in a third of the N operations, the third being performed after the second. The sequence of unitary operators to be applied in the third operation may be determined based on a comparison of at least the energy of the first quantum state to the energy of the second quantum state.

[0074] For example, if a comparison of the energy of a first quantum state with the energy of a second quantum state reveals that the energy of the first quantum state is less than the energy of the second quantum state, the user can conclude that the first quantum state is closer to the ground state of the measured Hamiltonian (e.g., the total Hamiltonian) than the second quantum state. Taking this into account, the user can reject the second sequence of unitary operators and return to the first sequence of unitary operations. Starting with the first sequence of unitary operations, the user can add a small perturbation to the sequence, for example, by replacing one or some of the operators in the sequence with another operator. The resulting sequence may be a sequence of unitary operators applied in a third operation.

[0075] Alternatively, if a comparison of the energy of the first quantum state with the energy of the second quantum state reveals that the energy of the first quantum state is greater than (or equal to) the energy of the second quantum state, the user can conclude that the second quantum state is closer to the ground state of the measured Hamiltonian than the first quantum state. With this in mind, the user can accept a second sequence of unitary operators. Starting with the second sequence of unitary operations, the user can make small adjustments or perturbations to the sequence. The resulting adjusted sequence can be a sequence of unitary operators applied in a third operation.

[0076] The user can proceed in a similar manner through all rounds of operation: (i) measure the energy of the quantum state obtained after applying the sequence of unitary operations of the current round (e.g., by measuring the total Hamiltonian); (ii) compare the measured energy of the current round with the measured energy of the previous round; (iii) if the measured energy of the current round is greater than the measured energy of the previous round, reject the quantum state of the current round and accept the sequence of unitary operations of the previous round, or if the measured energy of the current round is less than the measured energy of the previous round, accept the sequence of unitary operations of the current round; and (iv) starting from the accepted sequence of unitary operations, perturb the accepted sequence to obtain the sequence of operations for the next round of operation.

[0077] As the number of iterations, N, increases, a larger and larger set of quantum states is prepared, with the energy of subsequent quantum states gradually decreasing (or at least not increasing). Thus, the energy gradually approaches the ground state energy of the measured Hamiltonian. With this in mind, the embodiments described herein provide an increasingly improved approximation to the ground state of the measured Hamiltonian.

[0078] By measuring the appropriate Hamiltonian at each iteration of the operation, a solution to the computational problem can be determined. According to embodiments, multiple iterations of the N operations (specifically, substantially all of the N iterations) can each include measuring the total Hamiltonian of the quantum system. As described herein, the total Hamiltonian has a ground state that contains information about the solution to the computational problem. Thus, if the quantum system is at or near the ground state of the total Hamiltonian, measuring the quantum system can reveal information about the problem. The solution to the computational problem can be determined.

[0079] Although the above provides an example of how the measured Hamiltonian is the total Hamiltonian, other Hamiltonians can also be measured. Specifically, the total Hamiltonian may be modified or transformed such that the form of the modified Hamiltonian differs from the total Hamiltonian, but the modified Hamiltonian still has the property that the ground state of the modified Hamiltonian encodes the solution to the computational problem. For example, modifying the total Hamiltonian by changing the basis of each component or multiple small groups of components may change the form of the Hamiltonian, but does not change the property that the ground state of the modified Hamiltonian encodes the solution to the computational problem.

[0080] While the above provides examples of how one or more energies are measured, additional or alternative measurements may also be performed. For example, instead of measuring the energy of a quantum state, one may measure the quantum fidelity of the quantum state relative to a target state. For example, the target state may be a ground state of a target Hamiltonian (such as the total Hamiltonian described herein or another Hamiltonian). The quantum fidelity of two quantum states |ψ1〉 and |ψ2〉 refers to the quantity |ψ1|ψ2〉|.

[0081] Output of results

[0082] A method according to embodiments described herein includes outputting a result of a quantum computation. The result of the quantum computation can be based on one or more measurement results of one or more measurements performed in N operations. The method can include processing, for example, by a classical computing system described herein, one or more measurement results of the one or more measurements performed in the N operations. The method can include outputting a result of the quantum computation based on the one or more processed measurement results.

[0083] The output result of the quantum computation may be a solution to the computational problem, specifically a trial solution. The trial solution may be, for example, an approximate solution to the computational problem.

[0084] Subsystem

[0085] It may be beneficial to select a first subset S1 of summand constraint Hamiltonians in such a way that the summand constraint Hamiltonians in said first subset effectively define a division of the quantum system into smaller component groups called subsystems. This approach is also referred to herein as “modularization.”

[0086] 3 and 4 show schematic diagrams of a quantum system 300 including components 302. The components 302 may be spatially arranged along a two-dimensional lattice (hereinafter referred to as a "first two-dimensional lattice").

[0087] 3 shows the summand constraint Hamiltonians 320 of a first subset S1 of the summand constraint Hamiltonians (for ease of representation, the summand constraint Hamiltonians of the second subset S2 are not shown). The summand constraint Hamiltonians 320 are shown schematically as rectangles acting on groups of components (four-body operators in this example). The summand constraint Hamiltonians 320 are spatially arranged along a lattice pattern (a "second two-dimensional lattice," described below). In the illustrated example, the lattice pattern includes two horizontal lines and two vertical lines of the summand constraint Hamiltonians 320.

[0088] The spatial arrangement of the summand constraint Hamiltonian 320 can define a subdivision of the quantum system 300 into subsystems 450, with each subsystem 450 consisting of a subgroup of components 302, as shown in FIG. 4. Each subsystem 450 has a boundary formed by boundary components 420. Boundary components 420 are depicted in FIG. 4 as circles with crosses. Each boundary component 420 of a subsystem 450 participates in the summand constraint Hamiltonian 320, which jointly act on the subsystem 450 in question and on adjacent subsystems 450. The boundary components 420 of a subsystem 450 form the boundary between that subsystem and one or more adjacent subsystems 450.

[0089] A quantum system may contain subsystems. Each subsystem may contain a subset of the components of the quantum system. Subsystems may be decoupled. Different subsystems may have no components in common. Each subsystem may have boundary components that are part of, or form the boundary between, the subsystem and one or more adjacent subsystems.

[0090] Each boundary component can participate in a quantum interaction represented by an augend constraint Hamiltonian from a first subset S1 of the augend constraint Hamiltonians of the constraint Hamiltonians. A boundary component participates in a quantum interaction represented by an augend constraint Hamiltonian means that the augend constraint Hamiltonian acts on the boundary component in question. For each subsystem, each boundary component of the subsystem can be coupled to a boundary component of an adjacent subsystem by a quantum interaction represented by an augend constraint Hamiltonian of the first subset S1 of the augend constraint Hamiltonians.

[0091] Each subsystem may have a total number of components that is much smaller than the total number of components of the quantum system. For example, the total number of components of each subsystem may be 30% or less, 20% or less, 10% or less, or even 1% or less of the total number of components of the quantum system. Each subsystem may include a total number of components that is independent of the size of the computational problem. The total number of components of each subsystem may be a constant. In other words, the subdivision of the quantum system into subsystems by the spatial arrangement of the summand constraint Hamiltonian from the first subset may be such that the number of components of each subsystem is independent of the size of the computational problem and, therefore, is bounded by a constant that is independent of the total number of components of the entire quantum system.

[0092] According to some embodiments, each unitary driving operator can operate entirely within one of the subsystems of a quantum system. A unitary driving operator can only operate on components belonging to the same subsystem. When the size of each subsystem of a quantum system is relatively small compared to the size of the entire quantum system, a unitary operator operating entirely within one of the subsystems is a manageable object with reduced complexity. Specifically, the depth of the circuit required to implement such a unitary driving operator can be relatively small. For example, if the size of the subsystem (total number of components) is constant, i.e., independent of the size of the computational problem, a unitary driving operator operating entirely within the subsystem can be implemented by a quantum circuit of constant depth, i.e., a highly parallelized quantum circuit. This is advantageous because it requires only a small amount of computational (time) resources.

[0093] Each unitary driving operator is realizable or can be realized by a quantum circuit of a certain depth. The term "depth" as used in this disclosure refers to the concept of circuit depth for circuits of logic gates, known in the field of computer science. A circuit of quantum gates, i.e., unitary operators, that operate on a set of components of a quantum system is said to be parallelizable to depth D if the quantum gates in the circuit can be grouped into D layers (slices) of gates, and in each layer, no two quantum gates operate on the same component. In other words, within each layer, each component is operated on by at most one quantum gate. Depth is a measure of how parallelizable a circuit can be. Because gates within a layer operate on different components, operations within the same layer of the circuit can be executed in the same time step ("in parallel"). Thus, a circuit that is parallelizable to depth D can be executed in D time steps. For a more detailed explanation of the concept of depth, see WO 2020 / 156680, which is incorporated herein by reference.

[0094] The constant depth refers to a depth that is independent of the number of components of a quantum system. The constant depth can be much smaller than the number of components of a quantum system. For example, the constant depth can be 30% or less, specifically 20% or less, or more specifically 10% or less of the number of components of a quantum system. The methods described herein are used to solve computational problems of increasing size, thereby requiring quantum systems of increasing system size. According to embodiments, regardless of the size of the computational problem, each unitary driving operator can be realized by a quantum circuit of constant depth D. That is, unlike the number of components of a quantum system, the depth D does not increase as a function of the size of the computational problem but is bounded from above by a constant. For example, D can be at most 100.

[0095] The components of the quantum system may be arranged along or at least a portion of a first two-dimensional lattice, such as a two-dimensional rectangular lattice (such as component 302 shown in FIG. 3). The first two-dimensional lattice may include a plaquette. A plaquette, or elementary square, consists of four components of the quantum system spatially arranged along the square. Each summand of the constraint Hamiltonian may be a four-body operator acting on the plaquette or a three-body operator acting on a subset of three components in the plaquette.

[0096] The set formed by the boundary elements of all subsystems of the quantum system (e.g., boundary elements 420 shown in FIG. 4) may be arranged along a second two-dimensional lattice (or at least a portion of the second two-dimensional lattice), which may have a larger lattice spacing than the first two-dimensional lattice.

[0097] Example Run

[0098] Quantum systems and their components (such as qubits), as described herein, are physical entities. Specific implementations of quantum systems / components and the interactions involved in the methods described herein are briefly described below. Further details are provided in EP 3113084 and WO 2020 / 156680. However, the methods described herein can be practiced with other specific implementations of the physical entities and their interactions, and the exemplary implementations are not to be considered limiting.

[0099] The component can be a superconducting qubit, such as a transmon or a flux qubit. A supercurrent propagating clockwise and counterclockwise, respectively, in the primary superconducting loop can form the quantum basis states |1> and |0> of the superconducting qubit. Furthermore, a flux bias through the secondary superconducting loop can couple the quantum basis states |0> and |1>.

[0100] Σ k a k σ^ z (k) Simplicial Hamiltonians of the form can be realized by magnetic flux interacting with superconducting qubits. Constrained Hamiltonians, such as the plaquette Hamiltonian, can be realized using ancillary qubits, which can be placed inside each plaquette. K km σ^ z (k) σ^ z (m) Interactions between qubits of the form can be realized by an inductive coupling unit containing a superconducting quantum interference device. Applying an adjustable magnetic flux bias to the superconducting quantum interference device increases the interaction coefficient by a factor K km can be adjusted. The summand constraint Hamiltonian is H sr,p =C(σ z (1) +σ z (2) +σ z (3) +σ z (4) -2σ z (p) -1)2 This involves the imposition of an energy difference between the |0> and |1> quantum ground states, z (k) σ z (m) and simplex σ z (l) Only pairwise interactions of terms are included, where σ z (p) denotes the ancillary qubit. Alternatively, the constrained Hamiltonian can be realized without ancillary qubits, for example by using a three-island superconducting device as a transmon qubit.

[0101] Furthermore, the flux bias through the first superconducting loop of the superconducting qubit can be set so that the ground states |0> and |1> have the same energy, i.e., the energy difference between these ground states is zero. Furthermore, the flux bias through the second superconducting loop can couple the ground states |0> and |1>. Thus, hσ^ x (k) Hamiltonians of the form are realizable for superconducting qubits.

[0102] In the case of a superconducting charge qubit or a superconducting flux qubit, the CNOT operation can be realized using an additional capacitive element coupled to the two qubits. The interaction strength is adjusted by a magnetic or electric flux applied to the additional element. Alternatively, the two qubits are coupled to two modes of a Josephson ring modulator. The simplicial unitary operator exp(itσ^ x (k) ) or exp(itσ^ z (k) ) can be achieved using a controlled external magnetic or electric flux.

[0103] For a superconducting qubit, the qubit states |0> and |1> can be measured with high fidelity using a measurement device that includes multiple superconducting quantum interference devices, specifically, N hysteretic DC superconducting quantum interference devices and a latch of N RF superconducting quantum interference devices controlled by bias lines (the number of bias lines varies as √N).

[0104] Alternatively, the quantum system may be realized by systems such as trapped ions, ultracold atoms, impurities (such as NV centers) in solid crystals, quantum dots, etc. For background on how Hamiltonians, unitary operators and measurements can be implemented in such systems, see EP 3113084 and WO 2020 / 156680. As already mentioned above, these are exemplary implementations and should not be considered limiting.

[0105] Device

[0106] According to a further embodiment, an apparatus 500 for performing quantum computation is provided, as shown in FIG. 5 . The apparatus 500 includes a quantum system 300 comprising component 302. The apparatus 500 includes a classical computation system 550. The classical computation system 550 is configured to encode the computational problem 110 into a problem Hamiltonian for the quantum system 300. The problem Hamiltonian is a simplicial Hamiltonian that is a sum of summand problem Hamiltonians. The classical computation system 550 is configured to determine a constraint Hamiltonian for the quantum system 300, where the constraint Hamiltonian is a sum of summand constraint Hamiltonians. The ground state of the total Hamiltonian encodes a solution to the computational problem. The total Hamiltonian includes or is a sum of the problem Hamiltonian and the constraint Hamiltonians. The classical computation system 550 is configured to determine a first subset of summand constraint Hamiltonians of the constraint Hamiltonian and a second subset of summand constraint Hamiltonians of the constraint Hamiltonian. The apparatus 500 includes a quantum processing system including a unitary evolution device 530 and a measurement device 540. The quantum processing system is configured to perform N operations, where N≧2. Each operation includes evolving the quantum system 300 according to a sequence of unitary operators by the unitary evolution device 530. The sequence includes a problem-encoded unitary operator, a constraint-enforcing unitary operator, and a unitary driving operator. Each problem-encoded unitary operator is either a unitary time evolution operator of an augend problem Hamiltonian of the problem Hamiltonian or a unitary time evolution operator of a summation of augend problem Hamiltonians of the problem Hamiltonian. Each constraint forcing unitary operator is a unitary time evolution operator of an augend constraint Hamiltonian taken from a first subset of augend constraint Hamiltonians of the constraint Hamiltonian, or is a unitary time evolution operator of a summation of augend constraint Hamiltonians taken from said first subset. Each unitary driving operator is a unitary operator that commutes with all augend constraint Hamiltonians from a second subset of augend constraint Hamiltonians of the constraint Hamiltonian.Each time includes performing a measurement of one or more components 302, shown as portions 525 of the quantum system 300, by the measurement device 540. The classical computing system 550 is further configured to output a quantum computation result 590. The apparatus 500 may be configured to perform any aspect of the method according to embodiments described herein.

[0107] Each of the N operations may include preparing an initial quantum state, for example, by at least one of a unitary evolution device and a measurement device.

[0108] The apparatus may include a controller. The controller may include or be a classical computing system. The controller may be connected to the quantum processing system. The controller may be configured to instruct the unitary evolution device to evolve the quantum system according to a sequence of unitary operators for each of the N operations. The controller may be configured to instruct the measurement device to perform measurements of one or more components of the quantum system at each of the N operations. The classical computing system may be configured to receive a set of measurement results from the measurement device, the measurement results being obtained from measurements performed during one or more of the N operations. The classical computing system may be configured to determine a sequence of unitary operators to be performed in future ones of the N operations based on the received one or more measurement results. The classical computing system may be configured to determine a result of a quantum computation, such as a solution to a computational problem, based on the received one or more measurement results.

[0109] The unitary evolution device and the measurement device may be configured to perform any unitary operator and any measurement, respectively, as described in connection with the methods described herein. For example, the unitary evolution device may be configured to implement a quantum circuit comprising quantum gates for performing a unitary operation of a sequence of any of the N unitary operations. For example, the measurement device may be configured to perform any energy measurement, e.g., a measurement of the total Hamiltonian, as described herein, such as a measurement of the energy of the first quantum state and / or the second quantum state described herein.

[0110] Classical computing systems are distinguished from quantum computing systems. A classical computing system may be understood as a computing system that stores and processes information using only classical information carriers, such as classical bits. A classical computing system may not use quantum information carriers, such as qubits, to process information. A classical computing system may include a central processing unit (CPU) for processing information using classical bits and / or a memory for storing information using classical bits. A classical computing system may include one or more conventional computers, such as personal computers (PCs), and / or a network of conventional computers.

[0111] The classical computation system of the apparatus described herein can be configured to determine a driving Hamiltonian that is a sum of augend driving Hamiltonians, where the driving Hamiltonian commutes with every augend constraint Hamiltonian of the second subset of augend constraint Hamiltonians. Each unitary driving operator can be a unitary time evolution operator of an augend driving Hamiltonian of the driving Hamiltonian or can be a unitary time evolution operator of a sum of two or more augend driving Hamiltonians of the driving Hamiltonian.

[0112] The classical computing system may be configured to perform any of the classical computing operations described herein, such as feedforward of information in adaptive rounds of operations, comparison of measured energies to determine a sequence of unitary operations to be applied to future rounds of operations, etc.

[0113] Further Aspects 1 introduction

[0114] Quantum approximate optimization algorithms (QAOAs) are gate-based algorithms designed to solve combinatorial optimization problems on modern noisy quantum devices. In principle, these algorithms are applicable to general combinatorial optimization problems. However, the limited inter-qubit connectivity of quantum devices complicates their implementation and can adversely affect the performance of practical QAOAs. The parity architecture described below resolves the mismatch between the connectivity of the problem graph and the hardware graph by mapping the problem-defining interactions onto a simplex problem Hamiltonian while simultaneously constraining the expanded Hilbert space by a short-range constraint Hamiltonian. Specifically, starting from a classical compilation procedure that adjusts the qubit placement and constraints on the optimization problem, the parity architecture enables mapping general, i.e., long-range, highly connected, optimization problems onto a fixed qubit layout using only short-range quantum interactions.

[0115] One can consider a parity-architecture QAOA implementation where the constraints are explicitly enforced through an energy penalty. Alternatively, constraint preservation can be achieved implicitly by making the involved operators commute with the constraint Hamiltonian. While the first approach allows for a parallelizable implementation of QAOA with a reduced circuit depth, the latter approach significantly improves the probability of success.

[0116] Embodiments described herein provide a hybrid approach that reduces the number of explicitly enforced constraint terms, thereby improving performance, while keeping the required quantum circuit depth constant. This is achieved by dividing the constraint terms into a subset that is explicitly applied (a first subset of the summand constraint Hamiltonian described herein) and a subset where the constraints are implicitly preserved by adapting the driving Hamiltonian (a second subset of the summand constraint Hamiltonian described herein).

[0117] Figure 6 shows an example of the modularization of a parity-compiled computational problem, as described in more detail below. Figure 6(a) shows the problem graph to be executed. The problem graph includes nodes and edges between the nodes. Each node in the problem graph has a classical spin s i =±1 are placed. An edge between two nodes represents the presence of a (non-zero) interaction between the corresponding classical spins. A subgraph of the problem graph is shown, with nodes indicated by solid black dots and edges between the nodes. Figure 6(b) shows the quantum implementation layout of the parity coding problem. The circles represent components 302, more specifically parity qubits. In the example shown, each parity qubit corresponds to an edge in the problem graph shown in Figure 6(a), i.e., an interaction between two classical spins. The classical spin s i and s jIn the case of interactions between , the corresponding parity qubits are labeled with ij (as shown in Figure 6(c)). Squares and triangles, given with or without hatching, represent implicitly and explicitly enforced constraint terms, respectively. Squares represent four-regime constraint terms, and triangles represent three-regime constraint terms. Explicitly enforced constraint terms (hatched squares and triangles) can be used to partition a quantum system into modules or subsystems that can be treated separately. Explicitly enforced constraint terms are the summand constraint Hamiltonians 320 of the first subset S1 of summand constraint Hamiltonians, as described herein. Figure 6(c) shows an example module corresponding to the highlighted subgraph in Figure 6(a), with hybrid drive lines shown as solid and dashed lines connecting subsets of qubits. The modularization enables highly parallelizable circuit implementation of the required drive terms.

[0118] In other words, Figure 6(c) shows an example layout of parity qubits and constraints for the computational problem described by the subgraph shown in Figure 6(a). The layout in Figure 6(c) has constraints that are explicitly enforced (hatched triangles), while other constraints (open squares) are implicitly conserved by the drive and act simultaneously on one qubit in the indicated line. By selecting which constraints are in which subsets, we can partition the larger layout into smaller modules (see Figure 6(b)), allowing parallel execution of all required unitaries with an adjustable maximum circuit depth. 2 Parity QAOA

[0119] The task of finding a solution to a computational problem, such as a combinatorial optimization problem, can be formulated as the energy minimization of a general (classical) N-spin Hamiltonian function of the form

number

[0120] Due to the two-body and quasi-local nature of the physical interactions between components, such as qubits, of a quantum system, couplings such as those arising in Equation 1 are difficult, if not impossible, to implement directly in quantum hardware. In this disclosure, we utilize the parity mapping described in EP 3113084 and WO 2022 / 008057, which allows any high-order long-range k-body terms to be encoded into the Hamiltonian (the "total Hamiltonian" described herein) of a quantum system containing qubits (or, more generally, components) arranged on a square lattice. The total Hamiltonian includes only short-range interactions. This includes the problem spin s i Mapping a k-fold product of a subset of J onto a single qubit (herein called a “parity qubit” and denoted by σ̂), e.g., J ijl (3) s i s j s l →J m σ^ z (m) where we label each parity qubit with the corresponding k-tuple of problem spin indices; that is, in the given example, the index m is a contraction of the 3-tuple jjl. As a result, the K nonzero interaction terms in Equation 1 that define the computational problem are intensities J acting on each of the K parity qubits. m The corresponding simplicial Hamiltonian is referred to herein as the problem Hamiltonian, H^ P =Σ mJ m σ^ z (m) (Section J m σ^ z (m) is the summand problem Hamiltonian described herein. The Hilbert space formed by the K parity qubits, to which an ancillary qubit may be added, is parity The optimization problem is shown as H parity In order to ensure that the Hamiltonian is embedded in the low-energy subspace (basis space) of d Constraints C l is given. Therefore, H^ C =Σ l C^ l A constraint Hamiltonian of the form C^ is provided. l is referred to herein as the summand constraint Hamiltonian, or constraint term for short. This gives rise to a total Hamiltonian of the form

number

number

[0121] For further details regarding parity mapping, see EP 3113084 and WO 2022 / 008057. 2.1 Explicit Parity QAOA

[0122] The quantum approximate optimization algorithm (QAOA) is based on the driving Hamiltonian H^ B =Σ i N σ^ x (i) It is designed to find low-energy solutions to the quantum mechanical implementation H^ of the Hamiltonian function H in Eq. 1 by evolving the quantum state using alternating unitary time evolutions of and of a Hamiltonian H^ of variable period (H^ is the quantum state for each classical spin s in H). i σ^ z To perform QAOA in a parity architecture, we use the single-qubit driving Hamiltonian H^ X =Σ i K σ^ x (i) can be chosen in a similar way, while the Hamiltonian H^ is the total Hamiltonian H^ mentioned above. total =H^ P +H^ C See WO 2020 / 156680. Therefore, a parity QAOA sequence of depth p can be expressed as a Hamiltonian H^ X , H^ P and H^ C This corresponds to the variational evolution of a quantum system with

number

[0123] Figures 7a-c show an example of a parity-coded complete graph with six spin variables. The example problem graph (not shown) has six nodes, each with a spin variable s i = ±1 (i ranges from 0 to 5) and edges are given between every pair of nodes (the complete graph), i.e., an interaction exists between every pair of classical spins. The parity qubits are arranged on a two-dimensional lattice. The parity qubits are labeled ij, where i and j range from 0 to 5. A parity qubit ij has classical spin s i and s j The squares and triangles represent the four- and three-regime constraint terms (summand constraint Hamiltonian) of the constraint Hamiltonian, respectively. In Fig. 7a), all constraint terms are explicitly enforced (represented by hatched squares and triangles). The driving Hamiltonian includes a single qubit σ^ for all qubits. x Contains operators. 2.2 Implicit Parity QAOA

[0124] As mentioned above, -iΩH ^ C Instead of explicitly enforcing the constraint Hamiltonian in the QAOA protocol using a time-evolving unitary operator of the form

number

number

[0125] Figure 7(b) illustrates the concept of implicit parity QAOA. The displayed lines connecting certain subsets of qubits (e.g., subset {05, 15, 25, 35, 45}, subset {01, 12, 13, 14, 15}, etc.) indicate qubits that can be simultaneously flipped without leaving the constraint-satisfying subspace (constraint-preserving driving line). All constraints are implicitly enforced via the driving Hamiltonian, and no energy penalty is required for the constraint terms. For a complete graph such as that shown in Figures 7(a)–7(c), the set of qubits that can be simultaneously flipped without leaving the constraint-satisfying subspace is the set of qubits whose labels share a common problem spin index (e.g., the set or driving line consisting of qubits 01, 12, 13, 14, and 15 has the common label 1). Therefore, every set can be directly associated with one of the problem spins (e.g., the aforementioned driving line consisting of qubits 01, 12, 13, 14, and 15 can be associated with classical spin s1). In general, all sets of qubits that intersect each relevant constraint even times are valid choices. Hence, the σ̂ acting on these sets of qubits xFrom the product of these, we obtain the driving Hamiltonian H^ X imp , equation 6 is satisfied. In the example shown in Figure 7b), this is the parity-mapped analogue of the driving Hamiltonian acting on quantum mechanical problem spins. Every line in the diagram can be associated with exactly one problem spin, i.e., the spin whose index occurs in all parity qubits of the set. The spin variable s i is reflected in a flip of the quantum states of all parity qubits containing that index.

[0126] In general, the constraint-maintaining driving Hamiltonian H^ X imp The elements of can be defined as follows:

[0127] Which constraint term C^ l is satisfied (a constraint is "satisfied" meaning that the quantum system is in the ground state of the constraint), μ These qubits are typically arranged in a linear layout (see, for example, Figure 7b), or, more generally, appear as a tree graph of adjacent qubits. In what follows, we consider these sets Q μ is called the constraint-maintaining driving line, or simply the driving line.

[0128] The index μ lists the driving lines of the given computation problem. To each driving line we associate the following driving terms:

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[0129] In contrast to the QAOA approach that explicitly enforces all constraints, we examine the performance of parity QAOA that utilizes a constraint-maintaining drive Hamiltonian of the following form, composed of operators associated with a valid set of constraint-maintaining drive lines.

Number

[0130] In the example shown in Fig. 7b), the valid set of drive terms is given by D = {X^ (μ) |1 ≤ μ < N}, where X^ (μ) is shown by the following equation.

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[0131] Constraint-maintaining driven Hamiltonian H^ X imp is all the constraints C^ l Note that since , the number of violated constraints remains constant during the time evolution. Specifically, as long as the initial quantum state is prepared in a subspace that satisfies the constraints, the variational quantum state that results from the unitary time evolution is guaranteed to correspond to a valid configuration of the problem spin and thus to be a potential solution to the computational problem throughout the time evolution.

[0132] Constraint-maintaining driven Hamiltonian H^ X imp Using , we can prepare a variational QAOA state using the following protocol.

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[0133] Unitary time evolution e -iβH ^ Ximp There are practical drawbacks to this implementation: arbitrarily long drive lines and their overlaps can make parallel, low-depth execution of the corresponding gate sequences impossible, especially for large system sizes. Fully implicit implementations generally require a unitary time evolution e -iβH ^ Ximp To execute, we require a circuit depth that scales at least linearly with the system size (number of qubits). Because keeping the circuit depth small is a key point in many quantum computing settings, in the following, hybrid execution is considered as a way to balance between fully explicit and fully implicit execution of the constraint terms. In the hybrid approach, short driving lines are used to ensure parallelizable execution, reducing the required number of explicitly enforced constraints compared to fully explicit execution (see Figure 7c).

[0134] We start with a fully implicit implementation. Switching constraints from implicit to explicit implementation doubles the dimension of the reachable subspace. Therefore, the driving Hamiltonian can contain one additional term that does not commute with the problem constraints. Explicit enforcement, i.e., n enforced by a unitary time evolution operator, is C Consider a set of constraint terms (herein referred to as the first subset of the summand constraint Hamiltonian). The remaining constraint terms (herein referred to as the second subset of the summand constraint Hamiltonian) are implicitly enforced, i.e., by providing an appropriate driving Hamiltonian that preserves the constraints of the problem. In the hybrid Hilbert space H hybDefine ∑ as the space spanned by the computational basis states that satisfy all implicitly enforced constraint terms (i.e., all summand constraint Hamiltonians in the second subset). Note that

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[0135] The set D of hybrid driving lines is a Hilbert space H that is independent and satisfies the constraints CF A set D is valid if and only if any computational basis state in can be transformed into any other state by applying the operators associated with the driving lines of D. This definition is not as strict as the definition of validity for a complete constraint-preserving driving line. d ≦|D|≦N+n C -n d can include driving terms of exactly N+n C -n dThe inclusion of independent elements is a sufficient condition for a set of hybrid driving lines to be valid, but not a necessary condition. The set of all quantum states that satisfy the constraints, i.e., the Hilbert space H that satisfies the constraints, CF It is okay to include fewer lines if all quantum states in are still reachable (and these are the only quantum states that need to be reached). In what follows, |D|=N+n C -n d We focus on the case where, because some of the originally explicitly enforced constraints are naturally preserved by such driving, we can remap cases with fewer driving terms to this case by reevaluating the split of explicitly and implicitly enforced constraint terms.

[0136] A hybrid drive line is a set of qubits that can be simultaneously flipped without violating any constraints except those explicitly enforced. The relevant drive terms are defined analogously to Equation 7, and are not necessarily constrained by the space H CF does not preserve the hybrid Hilbert space H hyb Save.

[0137] N classical spin variables and n C is explicitly enforced n C tot In the computational problem involving a constraint Hamiltonian with a total count of constraint terms, the hybrid drive Hamiltonian is defined as the drive terms X^ associated with the valid set of hybrid drive lines. (μ) can be thought of as follows using

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[0138] The fully implicit and fully explicit approaches are C = 0 and n C =n C tot Note that this corresponds to the limiting case of the hybrid approach of

[0139] The hybrid QAOA protocol is similar to Equation 4, but with H^ X →H^ X hyb and H^ C →H^ C hyb is replaced as follows:

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[0140] Below we illustrate the power of this new flexibility with a complete graph example, and then show how it can be applied to arbitrary graphs. 3.1 Example: Complete graph

[0141] Consider again a layout implementing the all-to-all connectivity problem graph (the complete graph) shown in Figures 7a–7c. When a single qubit is flipped, at least one constraint is violated. By continuing to flip more qubits until all constraints are satisfied again, the minimal set of flipped qubits corresponds to a constraint-preserving driving line, as shown in Figure 7b. Figure 7c shows a hybrid approach. Only the row of three-regime constraint terms at the bottom of the lattice (shown as hatched triangles) is explicitly enforced, while the remaining terms (open squares) are implicitly enforced by restricting the dynamics through the hybrid driving Hamiltonian. By explicitly enforcing all three-regime constraint terms, as shown in Figure 7c, these three-regime constraint terms no longer need to be maintained by the driving terms. Therefore, we can find a valid set of hybrid driving lines, including short driving lines that preserve only the implicitly enforced four-regime constraint.

[0142] Alternatively, this structure can be understood as follows: In a hybrid setting, the original drive lines in Figure 7b) can still be used as hybrid drive lines, even if the lower constraints are explicitly enforced. However, because the lower constraints no longer restrict the form of the drive Hamiltonian, we can add one hybrid drive line for each explicitly enforced constraint. Any additional allowed states violate the corresponding constraint. Therefore, the only hybrid drive lines that can be added while keeping existing drive lines independent are those that invert the corresponding constraint values ​​and preserve all others. An example of such a hybrid drive line is the portion of the original drive line that ends with the corresponding constraint. While combining full drive lines with newly added partial drive lines is a good choice, we want to avoid long drive lines to keep the circuit depth low. Replacing each long drive line with the symmetric difference between itself and the corresponding partial drive line results in two short drive lines, as shown in Figure 7c), effectively "splitting" each long drive line in two. For a valid set of hybrid drive lines D, we can add drive lines Q. μ by its own symmetric difference to another line Q ν It is easy to see that substituting X^ yields another valid set D'. In the case of related driving terms, the symmetric difference of two driving lines corresponds to the product of the related driving terms, so this substitution results in X^ (μ) →X^ (μ) X^ (ν) Therefore, we arrive at the hybrid driving Hamiltonian

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[0143] Figures 8a)-8b) show examples of constraint term placements based on partitioning into three-body and four-body constraints. The computational problem can be energy minimization of a general form, including k-body interactions for any k, as shown in Equation 1. Each interaction term in Equation 1 is mapped to a parity qubit. As shown in Figures 8a)-8b), the parity qubits are labeled by tuples of the form 04, 135, etc. Here, parity qubit 04 corresponds to a two-body interaction between two classical spins s0 and s4 under the parity mapping, parity qubit 135 corresponds to a three-body interaction between three classical spins s1, s3, and s5, and so on. Figure 8a) shows a special case with only four-body constraint terms, where the driving lines are chosen to be strictly horizontal or vertical and can be trivially parallelized. Figure 8b) shows a general case including both three-body and four-body constraint terms. Explicitly implementing all three-body constraint terms also allows for parallel execution of all horizontal (vertical) lines. The driving line shown preserves all four constraints (unhatched squares), but the three constraints are H^ C(hatched triangles). The drive lines for the top row of parity qubits are omitted because they can be obtained by symmetric differences from the others. Qubits not involved in any of the drive lines shown are part of a single-qubit drive (not shown).

[0144] Compiling a general graph (or hypergraph) into a parity architecture can result in various arrangements of 3-body and 4-body constraints. In the case of only 4-body constraints, a hybrid driving Hamiltonian can be constructed that preserves all constraints from only horizontal and vertical lines, as seen in Figure 8a. This also applies to mixed 3-body and 4-body layouts, where all 3-body constraints are explicitly enforced (see Figure 8b). A possible strategy for partitioning the constraints is as follows: all 3-body constraints and all 4-body constraints required to connect them to the boundaries are explicitly enforced, and the remaining 4-body constraints are implicitly enforced by the driving. The driving circuit can be run in two steps, with all horizontal driving lines and all vertical driving lines running in parallel, respectively.

[0145] Further optimization can further reduce the number of explicitly enforced constraint terms. Some of the three-regime constraint terms are automatically preserved by the horizontal and vertical drive lines mentioned above. In other cases, small adjustments to the hybrid drive lines (e.g., adding a small set of additional qubits) may be sufficient to preserve even more of the three-regime constraint terms. An example of such optimization is shown in Figure 9.

[0146] Figure 9 shows an example of an optimized set of explicitly enforced constraints (hatched shapes) such that a hybrid driving line that preserves the remaining constraints (unhatched) can be implemented in a parallelizable quantum circuit of reasonable depth. The unhatched triangles indicate three-regime constraints that can be implicitly implemented without significantly increasing the required circuit depth. The hatched squares are four-regime constraints that remain explicitly enforced to connect adjacent explicitly enforced three-regime constraints to their boundaries, simplifying the driving line.

[0147] A more detailed description of how to select which constraints to explicitly or implicitly enforce and how to find a valid set of hybrid drivelines is provided in Appendix B. 3.3 Modularization

[0148] Using the procedures described in the previous section and in Appendix B, the average length of the hybrid drive lines (and therefore the depth of the QAOA circuit) can grow linearly with the device dimensions, i.e., the number of qubits in the quantum system. Here, we exploit the concept of implicitly and explicitly enforced constraints to impose an upper limit on the length of the hybrid drive lines, thereby limiting the depth of the drive circuit to a tunable constant while avoiding a fully explicit approach.

[0149] To limit the length of the hybrid drive line, the maximum spacing l max We introduce additional (usually equidistant) rows and columns of constraints that are explicitly enforced by max Examples of such rows and columns are shown for = 5. Figure 10 shows a larger layout modularization of qubits with additional explicitly enforced constraints (hatched squares and triangles) arranged in a grid.

[0150] A portion of the layout bounded by explicitly enforced constraints is called a module, also referred to herein as a "subsystem" of the quantum system. For example, in FIG. 10, a module or subsystem is a 5×5 qubit array. Explicitly enforced constraints (hatched squares and triangles) define the boundaries of each module. Because hybrid drive lines do not need to traverse rows or columns of explicitly enforced constraints, each module can be treated individually when constructing the hybrid drive line.

[0151] Furthermore, the length of the hybrid drive lines within a module is limited. In particular, if all three constraints within a module are explicitly enforced (i.e., if there are strictly vertical and horizontal lines only), the length of the hybrid drive lines must be at most l max becomes.

[0152] The quantum circuits that implement the unit time evolution of each driving term of each module can be executed simultaneously. Therefore, in this approach, the circuit depth of the driving Hamiltonian implementation is a user-defined quantity that can be selected according to the current needs. max Therefore, the circuit depth of the driving Hamiltonian implementation remains constant regardless of problem or device size.

[0153] Even in the more general case where some of the three constraints are implicitly enforced within a module, the problem of finding a suitable hybrid driveline reduces to a small, individual problem per module, and the length of the hybrid driveline is still approximately l max is.

[0154] Therefore, in Figure 10, all solid lines (extending horizontally) and all dashed lines (extending vertically) can be executed in parallel. The dotted lines (which may have both horizontal and vertical portions) arise from the implicitly enforced 3-body constraint, but their contribution to depth is small and can be partially parallelized with other steps. In each sub-module, "missing" driving lines are omitted because they can be obtained by symmetric differencing of other sub-modules.

[0155] e -iγH ^ P and e -iΩH ^ Chyb The execution of (15) will always be constant-depth. This can be seen from the parallelization procedure in WO 2020 / 156680. Further optimization of the gate sequence reveals that the circuit depth for constraint execution is bounded by 20. Combined with the above findings, this guarantees that a single cycle of the QAOA sequence according to Equation 15 for any size computational problem can be executed in a constant-depth quantum circuit. 4 Preparing the initial state

[0156] As the initial quantum state for the optimization procedure, the Hilbert space H parity , H hyb or H CF We can use the (negative) ground state of the driving Hamiltonian, which corresponds to an equal superposition of all computational states across (satisfying all implicitly enforced constraints). This depends on whether a fully explicit, hybrid, or fully implicit approach is used. In a purely explicit approach, this is easily achieved by preparing each physical qubit in an equal superposition of computational basis states, whereas in implicit, and especially hybrid, approaches, the preparation of the initial states becomes more difficult. We consider a general hybrid driving Hamiltonian, H^, which includes the valid set D of hybrid driving lines and also includes the limiting cases of fully implicit and explicit parity QAOA. X hyb The quantum state we want to create is H^ X hyb All driving terms and H^C hyb The resulting quantum state is a simultaneous eigenstate (eigenvalue + 1) with all constraints of . Because this quantum state is a stabilizer state, known methods for preparing stabilizer states can be used to construct a quantum circuit that generates this stabilizer state from a product quantum state. The resulting circuit can have a large circuit depth in architectures with limited connectivity.

[0157] In the following, we propose a procedure to prepare an initial quantum state with a low circuit depth that scales linearly with the module size. First, we define |D| = log2dim(H hyb ) driving qubit (satisfying all implicitly enforced constraints), which is (μ) This can be done by defining one qubit per drive line such that acts as the bit-flip operator for that drive qubit. In this diagram, the desired initial state corresponds to all drive qubits being in the |+> state.

[0158] Operator Z^ corresponding to the phase reversal operator of the driven qubit μ (μ) For this to be a valid construction, the newly defined operators must satisfy the following (anti)commutation relations when μ≠ν:

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[0159] The goal is that all driving qubits μ are X^ (μ) -The first step is to prepare a state in the eigenstate |+>. This is easy to prepare and satisfies the constraints, which also corresponds to all driving qubits being in the |↑> state. [×]K (Space H parity To prepare the desired state, we start from the quantum system H corresponding to a π / 2 rotation around the y axis of all the driving qubits. parity This is a continuous rotation e -iπ / 4X ^ (μ) and e -iπ / 4Z ^ (μ) can be decomposed into, and is therefore performed using the operators defined previously.

[0160] However, problems arise when a qubit is involved in multiple driving lines. z Performing an operation affects all drive lines involved and can result in unwanted crosstalk that must be avoided. Therefore, whenever possible, qubits that are not involved in other drive lines should be selected to perform the phase operation. This is possible in the fully implicit case, where qubits with index 0 each participate in only one drive line, as shown in Figure 7(b).

[0161] If this is not possible, drive line Q μ Z^ for qubit k in ∋k (μ) The operation is performed on the other driving line Q containing qubit k. ν All driving qubits associated with Z^ (ν)This can be done as long as it is in the eigenstate of and is not affected by rotation. [×]K In this case, all the driving qubits are in the Z^ eigenstate. This allows us to (μ) A sequence of drive rotations can be found such that for each rotation there is at least one qubit on the corresponding drive line that is not included in any other drive line or that only participates in other drive lines whose state has not yet been rotated.

[0162] This procedure allows the preparation of any desired superposition state for more general hybrid driving Hamiltonians. The circuit depth for state preparation scales similarly to the unitary implementation for time evolution under a single driving Hamiltonian. The exact procedure for any layout is given in Appendix C. 5 Numerical results 5.1 Depth scaling of quantum circuits

[0163] Figure 11 shows the quantum circuit depth (vertical axis) required to execute a single step of the QAOA protocol for a layout such as that shown in Figure 7. The horizontal axis represents the relative amount of explicit constraints. This can be modified by considering different partitions of the constraints into explicit and implicit execution (i.e., different partitions of the set of summand constraint Hamiltonians into first and second subsets, as described herein). The left side corresponds to fully implicit execution. As the amount of explicitly enforced constraints increases, the enforcement of three constraints is initially explicit one by one, and finally, at the point marked with a cross, all three constraints are explicitly enforced. Next, additional lattices of explicitly enforced constraints are introduced for modularization, and the lattice spacing is gradually reduced.

[0164] The relative amount of explicitly enforced constraints n C / n C totStarting from ∑ = 0, circuit depth increases linearly with system size. The large factor in scaling circuit depth is due to excessive overlap of driving lines in the fully implicit setup (Figure 7b), which makes parallel execution impossible. Circuit depth can be reduced by increasing the number of explicitly enforced 3-constraint conditions until all 3-constraint conditions are explicitly enforced at the point marked with a cross in the plot. Circuit depth scales linearly with system size, but with a much smaller a priori factor. Further improvement is possible through layout modularization. This slightly increases circuit depth by a fixed number due to the additional execution cost of the constraint Hamiltonian. Because all explicitly enforced constraints in a modularized grid can be executed in parallel, the increase in depth is independent of system size. However, it is now possible to further reduce the circuit depth required for the driving terms. When the grid is large enough, the relationship between the reachable circuit depth and the relative amount of explicitly enforced constraints becomes independent of system size. 5.2 QAOA performance for various protocols

[0165] To demonstrate the benefits of this new approach, we compare the performance of fully implicit, hybrid, and fully explicit QAOA approaches in a parity scheme. For a QAOA cycle with p=3, the QAOA parameters in the range [0, 2π) are reps = 100 times. Note that in the fully implicit approach, there is one less QAOA parameter per cycle because the constraint part has been removed. At each initialization, we find a local optimum using the following classical procedure: We perform random QAOA parameter updates. If the energy of the system decreases after a parameter update, the new parameters are updated with probability p accept If n = 1, it is accepted, otherwise it is accepted with a probability that decreases exponentially with the increase in energy due to the new parameters. This procedure is repeated until the target value converges. reps The lowest energy expectation value E=<ψ|H^ among the initializations phys|ψ> [see Eq. 4] is maintained for each step. After optimization, the residual energy E of the system state, defined as res Calculate.

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[0166] Clearly, the residual energy increases as the number of explicitly enforced constraints increases. This is related to the fact that the size of the feasible subspace also increases with the number of explicitly enforced constraints. Note that the simulations described in this section do not consider the effects of quantum noise, such as bit-flip errors and decoherence. 6 summary

[0167] In summary, we have shown how to improve the performance of parity QAOA by interpolating between a standard single-qubit driving Hamiltonian and a driving Hamiltonian tailored to the computational problem. The proposed hybrid approach maintains the parallelism of fully explicit parity QAOA while improving performance by reducing the search space. For a fixed hardware layout, the tradeoff between circuit depth and search space size can be dynamically changed by adjusting the sizes of the implicitly driven and explicitly interconnected submodules.

[0168] The ideas presented here can be easily implemented on arbitrary grid arrangements of qubits, which is necessary to address questions about the performance of practical QAOAs using modular layouts for large problem sizes that are inaccessible to classical simulations. Appendix A Decomposition of the driving term

[0169] Figure 13 shows the unitary operator e corresponding to the time evolution under the driving term (see Eq. 7). -iβX ^ (μ) CNOT gate and R x shows a possible decomposition into a rotation gate. A driving line containing n connected qubits can be implemented with a circuit depth of at most n+2. Note that there are many possible representations of this unitary operator as a circuit; that is, the qubits used for the rotation are freely chosen. We can exploit this freedom to minimize the circuit depth of a sequence of such operators. Appendix B Partition constraints and driving line search

[0170] Below, we outline a general algorithm for finding valid hybrid drive lines (hereafter simply referred to as lines). Depending on how we choose our constraints / qubits / drive lines (although in principle any choice would work), certain steps can be improved (solutions can be found faster, solutions require fewer explicitly enforced constraints, or solutions require a smaller circuit depth). Furthermore, line constraints can follow various criteria, but the algorithmic approach remains the same. As explained in Section 3.3, we start with all constraints implicitly enforced, apart from a grid of explicitly enforced constraints for modularity. We examine all modules (connected sets of implicitly enforced constraints, where connected means connected via diagonal / left / right / top / bottom adjacencies). · Examine all qubits in the module (all qubits in the constraint terms of that module). -If the qubit is not already part of a horizontal line, create a horizontal line through it. Keep adding qubits to one side of the line until the last added qubit is no longer within the new (yet-unreached) implicitly enforced 4-body constraint. Do this on both sides. -If there are any implicitly enforced constraints remaining in the module that are not maintained by the line (i.e., do not commute with the corresponding driving term), select one of these constraints and extend the line with a qubit that is on the same plaquette (2x2 square) as the selected constraint and adjacent to an existing line qubit, but is not yet part of the line. If no such qubit exists, proceed to the "failure module." -Repeat the above steps (select another unmaintained constraint and extend the line accordingly) until all implicitly enforced constraints are maintained by the line. -If the qubit is not already part of a vertical line, do the same procedure on the vertical line. From every set of connected lines (i.e., every line in such a set can be reached from every other line by simply moving along the line and switching between overlapping lines), remove the longest line until the remaining lines are independent (i.e., a symmetric difference of two or more lines does not create another line in the set). If any line exceeds pre-set limits, or if no initial state preparation algorithm is found (see next section), proceed to the "Failure Module." Pre-set limits include, but are not limited to, maximum line length, line branches / bends (i.e., whether and how deviations from a literal straight line are allowed), or the depth of the resulting circuit. If the total number of independent lines is less than the number of required lines, proceed to the "failed module." The number of required lines is equal to the number of implicitly enforced constraints in the module, n. imp,mod and the number of qubits in the module, K mod So, K mod -n imp,modIt can be calculated as: "Failure Module": Explicitly enforce an additional constraint on the module and start this module again. The additional constraint must be a 3-body constraint (including the qubit of the line leading to the failure, if a single line can be identified as leading to the failure) unless the failure module has an explicitly enforced 3-body constraint that is not adjacent to any other explicitly enforced constraint (which is the reason for the failure). In that case, explicitly enforce an additional constraint (3-body or 4-body) such that all explicitly enforced constraints are connected to the boundaries (through adjacent explicitly enforced constraints). Note that multiple constraints may be required to connect to the boundaries, and there may be multiple ways to connect the boundaries.

[0171] At the latest, (a) all three constraints are explicitly enforced and connected to the boundaries via explicitly enforced constraints, and (b) the length limit is greater than or equal to the maximum dimension of the module (e.g., if the module contains 3 × 4 qubits, the maximum line length is 4). This is true even if no branches or bends are allowed. If branches or bends are allowed, a valid, admissible line will be found sooner. Appendix C Instructions for preparing the initial quantum state

[0172] The following instructions are valid for fully implicit, fully explicit, and general hybrid drivelines, but are only necessary for true hybrids, as the other cases are trivial. First, all the drive lines Q μ Execution priority P determines the execution order. μ is assigned as follows: P μ =0 drive line contains at least one qubit that is not part of any other drive line. ·Priority P μ >0 for all drive lines Q μ contains at least one qubit, otherwise it has low priority P ν <P μ Line Qν exists only in Then, for each driving qubit λ, a priority P λ Apply the following rotations in descending order:

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[0173] The initial state |ψ0> can be prepared as follows:

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[0174] If this procedure does not allow priorities to be assigned to all lines, update the explicit / implicit grouping of constraints according to Appendix B and try again. At the latest, it will work if all three constraints are explicitly enforced. Figure 14 shows an example of two sets of connected driving lines in a submodule with assigned priorities. In Figure 14, only implicitly enforced constraints are shown. X: Missing line not in the driving Hamiltonian. 0-3: Priority P of the line that performs the rotation. μ (P μ (Starting from =3). Priority P μ The rotation of the line with R μ This can be done with the qubits marked with .

[0175] While the above is directed to embodiments, other and further embodiments may be devised without departing from the scope, as determined by the claims.

Claims

1. 1. A method of performing quantum computing, comprising: Providing a quantum system (300) including a component (302); encoding a computational problem (110) into a problem Hamiltonian (150) of said quantum system, said problem Hamiltonian being a simplex Hamiltonian that is a sum of summand problem Hamiltonians (152); determining a constraint Hamiltonian (250) for the quantum system, the constraint Hamiltonian being a summation of summand constraint Hamiltonians (252), a ground state of the total Hamiltonian encoding a solution to the computational problem, the total Hamiltonian comprising a summation of the problem Hamiltonian and the constraint Hamiltonians; A first subset (S １ ) and a second subset of the summand constraint Hamiltonians of the constraint Hamiltonian (S ２ ) and determining performing N (N≧2) operations, each of which comprises: A procedure for preparing an initial quantum state; a procedure for evolving the quantum system according to a sequence of unitary operators, the sequence including a problem-encoded unitary operator, a constraint-enforcing unitary operator, and a unitary driving operator, wherein each problem-encoded unitary operator is a unitary time evolution operator of an augend problem Hamiltonian of the problem Hamiltonian, or is a unitary time evolution operator of a summand problem Hamiltonian of the problem Hamiltonian, each constraint-enforcing unitary operator is a unitary time evolution operator of an augend constraint Hamiltonian taken from the first subset of the augend constraint Hamiltonian of the constraint Hamiltonian, or is a unitary time evolution operator of a summand constraint Hamiltonian taken from the first subset, and each unitary driving operator is a unitary operator that commutes with all augend constraint Hamiltonians from a second subset of augend constraint Hamiltonians of the constraint Hamiltonian; performing measurements of one or more components of the quantum system; outputting the result of said quantum computation (590); A method for performing quantum computing, comprising:

2. 2. The method of claim 1, wherein for each of the N operations, evolving the quantum system according to the sequence of unitary operators in the N operations comprises executing at least some of the unitary operators in the sequence with a quantum circuit comprising quantum gates.

3. 3. The method of performing quantum computing according to claim 1, wherein the quantum system comprises subsystems (450) each including a subset of the components, the subsystems being disjoint from one another, each subsystem having a boundary component (420) forming part of a boundary between the subsystem and one or more adjacent subsystems, each boundary component participating in a quantum interaction represented by an augend constraint Hamiltonian of the first subset of the augend constraint Hamiltonian of the constraint Hamiltonian.

4. 4. The method of claim 3, wherein each unitary driving operator operates entirely within one of the subsystems of the quantum system.

5. 5. The method for performing quantum computing according to claim 3, wherein each subsystem has a total number of components that is independent of the size of the computational problem.

6. 6. A method for performing quantum computing according to claim 3, wherein each unitary driving operator is realized by a quantum circuit of constant depth.

7. 7. A method for performing quantum computing according to claim 1, wherein the initial quantum state of at least some of the N times is a ground state of a partially constrained Hamiltonian that is the sum of all summand constrained Hamiltonians obtained from the second subset of summand constrained Hamiltonians.

8. determining a driving Hamiltonian that is a sum of the summand driving Hamiltonians; 8. A quantum computing method according to claim 1, wherein the driving Hamiltonian commutes with all summand constraint Hamiltonians of the second subset of the summand constraint Hamiltonian, and each unitary driving operator is a unitary time evolution operator of an augend driving Hamiltonian of the driving Hamiltonian, or a unitary time evolution operator of a summand driving Hamiltonian of two or more augend driving Hamiltonians of the driving Hamiltonian.

9. The sequence of unitary operators of at least some of the N operations is A where p≧3. １ A ２ ...A ｐ or at least a subsequence of said form, and each A ｉ is X ｉ Y ｉ Z ｉ is a product of the form X ｉ , Y ｉ and Z ｉ One of the operators is a problem-encoding unitary operator, and X ｉ , Y ｉ and Z ｉ Another one of the constraint-enforcing unitary operators is X ｉ , Y ｉ and Z ｉ 9. The method of claim 1, wherein a further one of the operators is a unitary driving operator.

10. 10. The quantum computing method of claim 1, wherein the N operations include one or more adaptation operations, and for each adaptation operation, a unitary operator in the sequence of unitary operators in the adaptation operation is determined based on at least one measurement result of a measurement performed in a previous operation of the N operations.

11. 11. The method of performing quantum computing according to claim 1, wherein the N operations include a first operation, wherein a first quantum state of the quantum system is obtained by evolving the quantum system according to the sequence of unitary operators of the first operation, and wherein performing the measurement in the first operation includes measuring the energy of the first quantum state.

12. the N operations include a second operation performed after the first operation, evolving the quantum system according to the sequence of unitary operators of the second operation results in a second quantum state of the quantum system, and performing the measurement a second time includes measuring the energy of the second quantum state; The method comprises: comparing the energy of the first quantum state to the energy of the second quantum state; determining the sequence of unitary operators to be applied in a third of the N operations, the third operation being performed after the second operation, and the sequence of unitary operators to be applied in the third operation being determined based on at least a comparison of the energy of the first quantum state and the energy of the second quantum state; 12. The method of claim 11, comprising:

13. The problem Hamiltonian is H^ Ｐ =Σ ｋ J ｋ σ^ ｚ （ｋ） and has the form σ ｚ （ｋ） is the Pauli operator of the k-th component of the quantum system, and each J ｋ is a coefficient, and each term J ｋ σ^ ｚ （ｋ） is the summand problem Hamiltonian, and / or the constraint Hamiltonian is Ĥ Ｃ =Σ ｌ C^ ｌ and each C^ ｌ is C^ ｌ = a ｌ Z^ ｌ +b ｌ I, and Z^ ｌ is Pauli σ ｚ is the tensor product of the operators, I is the identity operator, and a ｌ and b ｌ are coefficients, and each C^ ｌ 13. The method of claim 1, wherein: is an augend-constrained Hamiltonian.

14. Each unitary driving operator has the form exp(itĤ), where t is a coefficient and Ĥ is Σ ｊ b ｊ X^ ｊ where each b ｊ are coefficients, and each X^ ｊ is Pauli σ Ｘ Tensor product of operators or a single Pauli σ Ｘ is an operator, denoted as Σ ｊ The quantum computing method according to claim 1 , wherein denotes a sum of two or more terms or a single term.

15. An apparatus (500) for performing quantum computing, comprising: a quantum system (300) including a component (302); A classical computing system (550), comprising: configured to encode a computational problem (110) into a problem Hamiltonian (150) of said quantum system, said problem Hamiltonian being a simplex Hamiltonian that is a sum of summand problem Hamiltonians (152); configured to determine a constraint Hamiltonian (250) for the quantum system, the constraint Hamiltonian being a summation of summand constraint Hamiltonians (252), a ground state of the summation Hamiltonian encoding a solution to the computational problem, the summation Hamiltonian comprising a summation of the problem Hamiltonian and the constraint Hamiltonian; A first subset (S １ ) and a second subset of the summand constraint Hamiltonians of the constraint Hamiltonian (S ２ ) the classical computing system configured to determine A quantum processing system including a unitary evolution device (530) and a measurement device (540), the quantum processing system configured to perform N (N≧2) operations, each of which: a procedure for evolving the quantum system by the unitary evolution device according to a sequence of unitary operators, the sequence including a problem-encoded unitary operator, a constraint-enforcing unitary operator, and a unitary driving operator, each problem-encoded unitary operator being a unitary time evolution operator of an augend problem Hamiltonian of the problem Hamiltonian, or a unitary time evolution operator of a sum of the augend problem Hamiltonians of the problem Hamiltonian, and each constraint-enforcing unitary operator the evolving procedure, wherein an augend is a unitary time evolution operator of an augend constraint Hamiltonian taken from the first subset of the augend constraint Hamiltonians of the constraint Hamiltonian, or is a unitary time evolution operator of a summation of augend constraint Hamiltonians taken from the first subset, and each unitary driving operator is a unitary operator that commutes with all augend constraint Hamiltonians from a second subset of augend constraint Hamiltonians of the constraint Hamiltonian; performing a measurement of one or more components of the quantum system by the measurement device; the classical computing system is further configured to output a result of the quantum computing (590). Quantum computing device.

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