Quantum computing method and quantum computing 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 computation. The methods use a quantum system that includes 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. Based on the readout obtained from the measurements, a computational problem is solved. [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, e.g., 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 bits can be processed by evolving the quantum system. For example, a group of qubits in a quantum system can be coupled to each other according to a specified interaction. The quantum system can be developed to process the information carried by the quantum system in order to perform a computation, i.e., to solve a computational problem. Often, quantum computers are 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 in order to perform any quantum computation. Long-range interactions are interactions that couple qubits that are far away from each other in a quantum system. Such long-range interactions are 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 have the drawback that sequences of short-range interactions are inherently sequential, i.e., they cannot be parallelized, increasing the execution time of a quantum computation. 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] Yet 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 of performing a quantum computation is provided. The method includes providing a quantum system including a component. 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 sum of summand constraint Hamiltonians. A ground state of the total Hamiltonian encodes a solution to the computational problem. The total Hamiltonian includes a sum of the problem Hamiltonian and the constraint Hamiltonian. 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 preparing 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-enforced 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-enforced 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 a unitary time evolution operator of a summation of augend constraint Hamiltonians 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 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 is provided for performing quantum computation. 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 simplex 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 sum of summand constraint Hamiltonians. A ground state of the total Hamiltonian encodes a solution to the computational problem. The total Hamiltonian includes a sum of the problem Hamiltonian and the constraint Hamiltonian. 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 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 taken 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 taken 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 Hamiltonian. 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 which can be combined with the embodiments described herein are evident from the dependent claims, the description and the drawings. [Brief description 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 how to encode the computational problem into a problem Hamiltonian. [Diagram 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. [Diagram 3] 1 shows the spatial configuration of a first set of summand constraint Hamiltonians acting on a quantum system including components. [Figure 4] 1 illustrates the decomposition of a quantum system into subsystems based on the spatial arrangement of a first set of summand constraint Hamiltonians. [Diagram 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 give examples of configurations of summand-constrained Hamiltonians with division into three and four constraints. [Figure 9] We present an example of an optimized set of explicitly enforced constraints so that a hybrid driving line that preserves the remaining constraints can be implemented in a parallelizable quantum circuit of shallow depth. [Figure 10]We show a modularization of the layout of qubits in 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 Fig. 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 PREFERRED EMBODIMENTS

[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. The present 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 are described. The structures shown in the drawings are not necessarily drawn to scale and may include details that are drawn in an exaggerated manner to allow a better understanding of the embodiments.

[0014] The 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 computing 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, quantum systems are real-world objects. A quantum system includes components. Components are physical quantum entities themselves, and can be considered as smaller d-level quantum systems that jointly form a quantum system. Specifically, components of a quantum system can be qubits. Qubits should be understood as physical entities that realize 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 (a quantum state prepared at the beginning of the quantum computation) and a final quantum state (a quantum state that the quantum computation finally ends up in). The final quantum state can be or be approximated to a ground state of the Hamiltonian of the quantum system, such as the total quantum Hamiltonian described herein. A quantum system can evolve from an initial quantum state towards or to a ground state of the total quantum Hamiltonian by executing a sequence of unitary operators. Such an evolution is a real-world process, in particular 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 of a computation problem. Said information can be revealed by measuring the quantum system or a part of it, i.e. at least some of its components. The act of measurement is also a physical / technical process. By measuring, a readout of the quantum system can be obtained. A readout of a quantum system is a set of measurements obtained by measuring the components of the quantum system, which involves 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 10000. K can range from 100 to 10000, or from 100 to 100000, although K can be greater than 100000. 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 a time parameter. A unitary operator of the form exp(itH^) shall 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 a 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 a unitary time evolution exp(itH^) may include 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 t. The interactions may be turned off after the period t has elapsed.

[0019] In a realistic system, at least a small amount of noise is always present. Thus, 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 the 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 of performing a quantum computation is provided. The method includes providing a quantum system including a component. 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 sum 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 sum 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 time including providing an initial quantum state. Each round includes evolving the quantum system according to a sequence of unitary operators, the sequence including or consisting of a problem-encoded unitary operator, a constraint-enforced 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-enforced 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 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 of the quantum system. The method includes outputting a result of the quantum computation.

[0021] According to the embodiments described herein, the (a priori unknown) solution of the computational problem is encoded in the basis space of the total Hamiltonian. To determine said solution, the quantum system evolves towards the ground state of the total Hamiltonian by unitary evolution, more specifically by applying a sequence of unitary operators during 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 of the computational problem.

[0022] The total Hamiltonian can be the sum of two parts, namely, the problem Hamiltonian and the constraint Hamiltonian. Thus, the ground state of the total Hamiltonian is characterized as a quantum state that has low energy with respect to both the problem Hamiltonian and the constraint Hamiltonian. Thus, by lowering the energy of the quantum system with respect to both the problem Hamiltonian and the constraint Hamiltonian, the quantum system can be evolved towards the ground state of the total Hamiltonian. By applying the problem-encoded unitary operator to a sequence of N unitary operators (the problem-encoded unitary operator is the time evolution of the (sum of) the summand problem Hamiltonian), the quantum system evolves into a region of states that have low energy with respect to the problem Hamiltonian. With respect to the constraint Hamiltonian, the summand constraint Hamiltonian is divided into two groups, namely, 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-enforced unitary operators) of (the summand constraint 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 of 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 a 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 constrained Hamiltonians are explicitly enforced and others are implicitly enforced. The explicit enforcement of the summand constrained 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 executed with a small circuit depth, making the practical realization significantly easier. Nevertheless, not all of the summand constrained Hamiltonians of the constraint Hamiltonians are explicitly enforced, since explicit enforcement would lead to an increase in the size of the subspace of quantum states searched in the course of the above iterative process, and thus an increase in the execution time of the computation. In contrast, the implicit enforcement of the second subset S2 of the summand constrained Hamiltonians, by its very structure, enforces the quantum system to remain within the basis space of the summand constrained Hamiltonians of the second subset S2, thereby limiting the size of the subdomain of the quantum system explored 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 may be understood as a measure of the number of classical information units, e.g., classical bits, required to specify a computational problem. The size of a computational problem may 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 may 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 may 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 that acts only on the kth component of the quantum system.

[0030] The problem Hamiltonian may have adjustable parameters. The adjustable parameters of the problem Hamiltonian may 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 may be a field, in particular a simplicial field. A simplicial field may refer to a field that affects a single component of the quantum system. The external entity may 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 adjustable parameters of the problem Hamiltonian may 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 the 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 Hamiltonian H^ P =Σ k H^ P,k Here we show the computational problem 110, which can be coded as P and each summand problem Hamiltonian H^ P,k are shown in FIG. 1 by the reference numerals 150 and 152, respectively.

[0033] Encoding the computational problem into the problem Hamiltonian may include determining, from the computational problem, a problem encoding configuration of the 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 configuration may be a set of parameters J that encode the computational problem to be solved by quantum computing. k may be a particular set of values.

[0034] Encoding a computational problem into a problem Hamiltonian can include 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 may include determining the problem Hamiltonian from an auxiliary computational problem. Specifically, the problem encoding configuration of the adjustable parameters of the problem Hamiltonian may be determined from the auxiliary computational problem. For example, each interaction between spins in the spin model of the auxiliary computational problem may be mapped to an augend problem Hamiltonian of the problem Hamiltonian. A specific encoding (called "parity" encoding) that allows 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.

[0037] Constraint Hamiltonian

[0038] The act of determining the constraint Hamiltonian may include determining a classical description of the constraint Hamiltonian. Determining may 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 may be understood similarly.

[0039] The constraint Hamiltonian (H^ C (denoted by ) 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 interactions occur between components that are separated from each other by a distance greater than the interaction cutoff distance D SR may be a constant distance. The interaction cutoff distance D SRcan be much smaller than the maximum component distance between components in the 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. If the 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 from 1 to 5. For example, r can be √2, 2, 3, 4, or 5. The short-range Hamiltonian H^ can be 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 We act only within groups of components of a quantum system that are separated from each other by a distance of 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, specifically 4 or less. A d-body Hamiltonian may refer to a Hamiltonian that represents interactions between multiple components, where there are no bonding interactions between groups containing d+1 or more components. A d-body Hamiltonian is a summation of augend Hamiltonians, where each augend Hamiltonian represents the bonding interactions between groups of d or less components.

[0041] The Z-type operator is 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 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, Z^ l is Pauli σ^ z is the tensor product of 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] In this specification, specific forms of the Hamiltonian (problem Hamiltonian, constraint Hamiltonian, driving Hamiltonian, etc.) are provided 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 one can freely choose the corresponding directions (x, y, z) or reorder the types of Pauli operators. 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 H^) has the property that the ground state of the Hamiltonian encodes the solution to a computational problem. Here, the total Hamiltonian is the problem Hamiltonian H^ P and the constraint Hamiltonian H^ C That is, 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 said 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 said measurements, a solution (e.g., a trial solution) to the computational problem can be determined.

[0044] The term "constraint Hamiltonian" comes from the property that the encoding of 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 may 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 the spin configurations of the Ising model, i.e., quantum states that cannot be "mapped back" to the Ising model. To remove these additional degrees of freedom, a constraint Hamiltonian is introduced. The constraint Hamiltonian imposes an energy penalty or energy constraint on said quantum states such 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 the solution of the computational problem. Specifically, each summand constraint Hamiltonian may impose a parity constraint on a subgroup of components such that the number of components in quantum state |1> within the subgroup of components is even.

[0045] Particular encodings that allow determining the problem Hamiltonian and the corresponding constraint Hamiltonians from an Ising spin model problem are described in EP 3113084 and WO 2022 / 008057.

[0046] As mentioned above, the total Hamiltonian may be the sum of the problem Hamiltonian and the constraint Hamiltonian. In other embodiments, the total Hamiltonian may include additional terms, in other words, the total Hamiltonian may include any additional terms in addition to the sum of the problem Hamiltonian and the constraint Hamiltonian. The additional terms may correspond, for example, to additional conditions ("side conditions") imposed on the solution of 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 shown 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 is made up of summand constraint Hamiltonians C^1, C^2, and C^3, and the second subset S2 is made up 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 augend constraint Hamiltonians of the constraint Hamiltonian are denoted herein as S1 and S2, respectively. The first and second subsets S1 and S2 may be disjoint subsets. The union of the first and second subsets S1 and S2 may form a whole set of augend constraint Hamiltonians of the constraint Hamiltonian. The first and second subsets S1 and S2 may form a partition of the augend constraint Hamiltonians of the constraint Hamiltonian. Specific examples of the first and second subsets of augend 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.

[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] At least some, and possibly all, of the initial quantum states 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 taken from the second subset S2 (the partially constrained Hamiltonian is the "second partially constrained Hamiltonian" as further described below). The partially constrained Hamiltonian has a basis space. The basis space is composed 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, that consists 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). At least some, and possibly all, of the initial quantum states of the N operations may be a superposition of all quantum basis states of the basis 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, which include 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 an augend problem Hamiltonian of the problem Hamiltonian, or a unitary time evolution operator of a sum of augend problem Hamiltonians of the problem Hamiltonian. A problem-encoded unitary operator may have the form exp(itA^), where t is a coefficient and A^ is a single augend problem Hamiltonian H^. P,k or two or more summands problem Hamiltonian H^ P,k (A^ is equal to the sum of all the summand Hamiltonians H^P,k Since A^ is the sum of H^ P (including cases where 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 a unitary time evolution operator of a sum of the 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 a base spatial basis of the second partially constrained Hamiltonian, where an operator is considered to act trivially in a base spatial basis if the operator maps each quantum basis state of the base 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 (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 basis spatial basis of the second partially constrained Hamiltonian. An operator is considered to have a nontrivial action in the basis spatial basis if the operator does not act trivially in the basis spatial basis. For each quantum basis state |x> in the basis 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 of the basis spatial basis of the second partially constrained Hamiltonian.

[0059] Methods according to embodiments described herein may include determining a driving Hamiltonian that is a sum of augend driving Hamiltonians, where the driving Hamiltonian commutes with all augend constraint Hamiltonians of the second subset S2. Each unitary driving operator may be a unitary time evolution operator of the augend driving Hamiltonian of the driving Hamiltonian, or may be a unitary time evolution operator of the sum of two or more augend driving Hamiltonians of the driving Hamiltonian (including the case where the unitary driving operator is a unitary time evolution operator of the driving Hamiltonian and the driving Hamiltonian is the 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 one can freely choose the corresponding directions (x, y, z) or permute the types of Pauli operators. 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 is 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, and possibly all, of U^1, U^2, . . . , U^2 may be problem-encoded unitary operators, constraint-enforced 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 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 or circuit of multiple unitary operators (quantum gates). This is the case for 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 several 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-driven 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 the quantum circuit may be a short-range unitary operator. A short-range unitary operator is a unitary operator that operates only within a subgroup of components of a quantum system, where any two components in the subgroup are separated from each other by a distance equal to or less than the interaction cutoff distance of the quantum system, as described herein. A short-range unitary operator does not operate on components outside of said subgroup of components.

[0065] Additionally or alternatively, each quantum gate of a quantum circuit may be a d-body 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-body unitary operator refers to a unitary operator that operates only within a subgroup that contains at most d components of a quantum system. A d-body unitary operator does not operate on components outside of said subgroup. A simplicial unitary operator is a d-body unitary operator with d=1.

[0066] The sequence of unitary operators for at least some of the N operations may include K problem-encoded 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 more, specifically 10 or more, and more specifically 200 or more.

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

[0068] The sequence of at least some unitary operators in N operations is A1A2···A p where p≧3, and in particular p is 10 or more, 100 or more, or 1000 or more. i 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-encoded 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 the above 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 a sequence of unitary operators for the operation. Performing the measurement of the one or more components may include performing a Pauli operator, e.g., a Pauli operator σ zThis may include measuring the

[0070] In some embodiments, the method includes information feed-forward, whereby 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, such as at least two previous operations. The N operations may include one or more adaptive operations, such as 10 or more, 100 or more, 1000 or more, or even 100000 or more adaptive operations. For each adaptive operation, a unitary operator of the sequence of unitary operators of 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 round operation. Evolving the quantum system according to the sequence of unitary operators in the first round operation can result in a first quantum state of the quantum system. Performing a measurement in the first round can include measuring an 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 an 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 the first quantum state to an energy of the second quantum state. The methods may 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 may be determined based on at least a comparison of 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 the first quantum state with the energy of the second quantum state reveals that the energy of the first quantum state is smaller than that of the second quantum state, the user may 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. In view of this, the user may reject the sequence of unitary operators in the second round and return to the sequence of unitary operations in the first round. Starting with the sequence of unitary operations in the first round, the user may add a small perturbation to the sequence, for example by replacing one or some operators in the sequence with another operator. The resulting sequence may be a sequence of unitary operators applied in a third round.

[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 may 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 may accept a second sequence of unitary operators. Starting with a second sequence of unitary operations, the user may make small adjustments or perturbations to said sequence. The resulting adjusted sequence may be a sequence of unitary operators to be applied in a third operation.

[0076] The user can proceed in a similar manner through all the rounds of operations: (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) reject the quantum state of the current round and accept the sequence of unitary operations of the previous round if the measured energy of the current round is greater than the measured energy of the previous round, or accept the sequence of unitary operations of the current round if the measured energy of the current round is less than the measured energy of the previous round; (iv) starting from the accepted sequence of unitary operations, perturb the accepted sequence to obtain a sequence of operations for the next round of operations.

[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 an embodiment, multiple iterations of the N iterations (specifically, substantially all of the N iterations) can each include measuring a 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 a ground state of the total Hamiltonian, measuring the quantum system may reveal information about the problem. A solution to the computational problem can be determined.

[0079] Although the above shows 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 is different 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. Changing the total Hamiltonian, for example 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] Although the above provides examples of how one or more energies are measured, further or alternative measurements may 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 one or more measurement results of one or more measurements performed in N operations, for example by a classical computing system described herein. 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 approximation of the computational problem.

[0084] Subsystem

[0085] It may be beneficial to select a first subset S1 of summand constrained Hamiltonians in such a way that the summand constrained 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 that includes components 302. The components 302 may be spatially arranged along a two-dimensional lattice (hereinafter referred to as a "first two-dimensional lattice").

[0087] In Fig. 3, the augend constrained Hamiltonians 320 of a first subset S1 of the augend constrained Hamiltonians are shown (for ease of representation, the augend constrained Hamiltonians of the second subset S2 are not shown). The augend constrained Hamiltonians 320 are shown diagrammatically as rectangles acting on a group of components (four-body operators in this example). The augend constrained Hamiltonians 320 are spatially arranged along a lattice pattern (a "second two-dimensional lattice" as described below). In the illustrated example, the lattice pattern includes two horizontal and two vertical lines of the augend constrained 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, which are shown in FIG. 4 as circles with crosses. Each boundary component 420 of a subsystem 450 participates in an summand constraint Hamiltonian 320 that jointly acts on the subsystem 450 in question and on adjacent subsystems 450. The boundary components 420 of a subsystem 450 form a boundary between that subsystem and one or more adjacent subsystems 450.

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

[0090] Each boundary component may participate in a quantum interaction represented by an augend constraint Hamiltonian from a first subset S1 of the augend constraint Hamiltonians of the constraint Hamiltonian. A boundary component participates in a quantum interaction represented by an augend constraint Hamiltonian means that said augend constraint Hamiltonian acts on the boundary component in question. For each subsystem, each boundary component of the subsystem may 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, or 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 limited by a constant that does not depend on the size of the computational problem and is therefore independent of the total number of components of the entire quantum system.

[0092] According to some embodiments, each unitary driving operator can act entirely within one of the subsystems of the quantum system. A unitary driving operator can only act on components belonging to the same subsystem. If the size of each subsystem of a quantum system is relatively small compared to the size of the entire quantum system, a unitary operator acting entirely within one of the subsystems is a manageable object of reduced complexity. In particular, the depth of the circuitry required to execute such a unitary driving operator may be relatively small. For example, if the size of the subsystems (total number of components) is constant, i.e. does not depend on the size of the computational problem, a unitary driving operator acting entirely within said subsystem can be executed by a quantum circuit of constant depth, i.e. a highly parallelized quantum circuit. This is advantageous since 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 of a circuit of logic gates known in the field of computer science. A circuit of quantum gates, i.e., unitary operators, that act on a set of components of a quantum system is said to be parallelizable to a depth D if the quantum gates in the circuit can be grouped into D layers (slices) of gates, and in each layer, there are no two quantum gates that act on the same component. In other words, within each layer, each component is acted on by at most one quantum gate. Depth is a measure of how far a circuit can be parallelized. Because gates in one layer act 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 a depth D can be executed in D time steps. For a more detailed explanation of the concept of depth, see WO 2020 / 156680. WO 2020 / 156680 is incorporated herein.

[0094] The constant depth refers to a depth that does not depend on the number of components of the quantum system. The constant depth can be a depth that is much smaller than the number of components of the quantum system. For example, the constant depth can be a depth of 30% or less, specifically 20% or less, more specifically 10% or less, of the number of components of the quantum system. The methods described herein are used to solve computational problems of increasing size, thus requiring quantum systems of increasing system size. According to the embodiment, 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 the quantum system, the depth D does not increase as a function of the size of the computational problem, but is limited 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 components 302 shown in FIG. 3). The first two-dimensional lattice may include a plaquette. A plaquette, i.e., a basic square, is composed 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 the 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). The second two-dimensional lattice may have a larger lattice spacing than the first two-dimensional lattice.

[0097] Example Run

[0098] Quantum systems and their components (such as qubits) are physical entities as described herein. Specific implementations of quantum systems / components and 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 performed on other specific implementations of said physical entities and their interactions, and the exemplary implementations are not considered limiting.

[0099] The components can be superconducting qubits, such as transmons or flux qubits. 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, for example the plaquette Hamiltonian, can be realized using ancillary qubits, which can be placed inside each plaquette. K km σ^ z (k) σ^ z (m) can be realized by an inductive coupling unit that includes a superconducting quantum interference device. Applying a tunable magnetic flux bias to the superconducting quantum interference device reduces the interaction 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 can be achieved by using the form σ 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 with 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 in solid crystals (such as NV centers), 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, as shown in FIG. 5, an apparatus 500 for performing quantum computation is provided. The apparatus 500 includes a quantum system 300 with a 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 simplex 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. A 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 Hamiltonian. 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-forcing 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 summand problem Hamiltonian 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 perform a quantum circuit comprising quantum gates for performing a unitary operation of a sequence of any of the N operations of 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 as 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 media, 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 summation 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 may be a unitary time evolution operator of an augend driving Hamiltonian of the driving Hamiltonian, or may be a unitary time evolution operator of a summation 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 of the methods described herein, such as feeding forward information in an adaptation round of operations, comparing 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 Approximation Optimization Algorithms (QAOA) are gate-based algorithms designed to solve combinatorial optimization problems on modern noisy quantum devices. The algorithms are in principle applicable to general combinatorial optimization problems. However, the limited inter-qubit connectivity of quantum devices complicates their implementation and may adversely affect the performance of QAOA in practice. 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-distance constraint Hamiltonian. Specifically, starting from a classical compilation procedure that adjusts the placement of qubits and constraints on the optimization problem, the parity architecture enables mapping general, i.e., long-range and highly connected, optimization problems onto a fixed qubit layout utilizing only short-range quantum interactions.

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

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

[0117] Figure 6 shows an example of a modularization of a parity-compiled computation problem, as described in more detail below. Figure 6(a) shows a 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. 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 shown as filled black dots, with edges between the nodes. Figure 6(b) shows a quantum implementation layout of the parity encoding problem. The circles represent components 302, more specifically parity qubits. In the example shown, each parity qubit corresponds to an edge of the problem graph shown in Figure 6(a), i.e. an interaction between two classical spins. The classical spin s i ands jIn the case of interactions between , the corresponding parity qubits are labeled with ij (as shown in FIG. 6(c)). Squares and triangles, given unhatched or hatched, represent implicitly and explicitly enforced constraint terms, respectively. Squares represent four-regime constraint terms, and triangles represent three-regime constraint terms. The explicitly enforced constraint terms (hatched squares and triangles) can be used to partition the quantum system into modules or subsystems that can be treated separately. The explicitly enforced constraint terms are the summand constraint Hamiltonians 320 of the first subset S1 of summand constraint Hamiltonians, as described herein. FIG. 6(c) shows an example module corresponding to the highlighted subgraph of FIG. 6(a), with hybrid drive lines shown as solid and dashed lines connecting the subsets of qubits. The modularization allows for highly parallelizable circuit implementation of the required drive terms.

[0118] In other words, Fig. 6(c) shows an example layout of parity qubits and constraints for the computation problem described by the subgraph shown in Fig. 6(a). In the layout of Fig. 6(c), some constraints are explicitly enforced (hatched triangles) while others (open squares) are implicitly preserved by the drive and act simultaneously on one qubit in the line shown. By selecting which constraints are included in which subsets, a larger layout can be partitioned into smaller modules (see Fig. 6(b)), allowing parallel execution of all required unitaries with tunable 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 occurring in Equation 1 are difficult, if not impossible, to implement directly in quantum hardware. In this disclosure, we exploit the parity mapping described in EP 3113084 and WO 2022 / 008057, which allows any higher-order long-range k-body terms to be encoded into the Hamiltonian (the "total Hamiltonian" described herein) of a quantum system that includes 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 to a single qubit (herein called a “parity qubit” and denoted by σ̂), e.g., 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 for the 3-tuple jjl. As a result, the K nonzero interaction terms in Eq. 1 that define the computation problem are expressed as intensities J 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 ancillary qubits may be added, is given by H parity The optimization problem is denoted by 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 a method for solving the problem of B =Σ i N σ^ x (i) It is designed to find low energy solutions for 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 a Hamiltonian H^ of variable period (H^ is the classical spin s of H). i σ^ z To perform QAOA in a parity architecture, we need to 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. Thus, a parity QAOA sequence of depth p can be expressed as the Hamiltonian H^ X , H^ P and H^ C This corresponds to the variational evolution of a quantum system with

number

[0123] 7a)-7c) 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 (with i ranging 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, with i and j ranging from 0 to 5. A parity qubit ij has classical spin s i ands j The squares and triangles represent the four- and three-regime constraint terms of the constraint Hamiltonian (summand constraint Hamiltonian), respectively. In Fig. 7a), all constraint terms are explicitly enforced (represented by hatched squares and triangles). The driving Hamiltonian contains 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

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[0125] Figure 7b) 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 the qubits that can be simultaneously flipped without leaving the subspace that satisfies the constraints (constraint-preserving driving lines). All constraints are implicitly enforced via the driving Hamiltonian, and no energy penalty for the constraint terms is necessary. For a complete graph such as that shown in Figures 7a)-7c, the set of qubits that can be simultaneously flipped without leaving the subspace that satisfies the constraints 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, 15 has a 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, 15 can be associated with classical spin s1). In general, all sets of qubits that intersect with each relevant constraint an even number of times are valid choices. xFrom the product of these, we obtain the driving Hamiltonian H^ X imp Constructing the spin variable s satisfies equation 6. In the example shown in Fig. 7b), this is the parity-mapped analogue of the driving Hamiltonian acting on the quantum mechanical problem spin. Every line in the diagram can be associated with exactly one problem spin, i.e., a spin whose index occurs in all parity qubits of the set. i is reflected in an inversion of the quantum states of all parity qubits that contain 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 A set of qubits Q that can be flipped simultaneously without changing whether μ These qubits are typically arranged in a line-like layout (see, for example, Fig. 7b) or, in the more general case, appear as a tree graph of adjacent qubits. In what follows, we call these sets Q μ is called the constraint-preserving driving line, or driving line for short.

[0128] The index μ enumerates the driving lines of a given computational 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, which is composed of operators associated with a valid set of constraint-maintaining drive lines.

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[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 represented 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 �� commutes with ��, 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, it is guaranteed that the variational quantum state resulting from the unitary time evolution corresponds to a valid configuration of the problem spin and is therefore a potential solution to the computational problem throughout the time evolution.

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

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[0133] Unitary time evolution e -iβH ^ Ximp There are practical drawbacks to the implementation: arbitrarily long drive lines and their overlaps may preclude parallel, low-depth execution of the corresponding gate sequences, especially for large system sizes. Fully implicit implementations generally do not support a unitary time evolution e -iβH ^ Ximp In order to execute, we need a circuit depth that scales at least linearly with the system size (number of qubits). Since keeping the circuit depth small is a key point in many quantum computing settings, in the following a hybrid execution is considered as a way to balance between a fully explicit and a fully implicit execution of the constraint terms. In the hybrid approach, short driving lines are used to ensure a parallelizable execution, reducing the required number of explicitly enforced constraints compared to a fully explicit execution (see Fig. 7c).

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

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[0135] A set D of hybrid driving lines is a Hilbert space H that satisfies the constraints and is independent. 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 line 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. We define all quantum states that satisfy the constraints, i.e., the Hilbert space H CF It is OK to have 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,Since some of the originally explicitly enforced constraints are naturally preserved by such driving, we can remap the case 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 for 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 C tot In the computation 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 a valid set of hybrid drive lines. (μ) can be thought of as follows:

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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, resulting in the following:

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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 (complete graph) as shown in Fig. 7a)-7c). When a single qubit is flipped, at least one constraint is violated. If we continue to flip more qubits until all constraints are satisfied again, the minimal set of flipped qubits corresponds to a constraint-preserving driving line, as in Fig. 7b). Fig. 7c) shows a hybrid approach. Only the row of 3-constraint terms at the bottom of the lattice (shown as hatched triangles) is explicitly enforced, while the other terms (open squares) are implicitly enforced by restricting the dynamics by the hybrid driving Hamiltonian. As shown in Fig. 7c), explicitly enforcing all 3-constraint terms removes the need for these 3-constraint terms to be maintained by the driving terms. Thus, we can find a valid set of hybrid driving lines, including short driving lines that preserve only the implicitly enforced 4-constraint terms.

[0142] Alternatively, this structure can be understood as follows. In the hybrid setting, the original drive lines in Fig. 7b) can still be used as hybrid drive lines, even if the lower constraint terms are explicitly enforced. However, because the lower constraint terms no longer restrict the form of the drive Hamiltonian, we can add one hybrid drive line for each explicitly enforced constraint term. Any additionally allowed states violate the corresponding constraint term. Thus, the only hybrid drive lines that can be added while keeping the 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 part of the original drive line that ends with the corresponding constraint term. While combining the complete drive line with the newly added partial drive line is a good choice, we want to avoid long drive lines to keep the circuit depth low. Replacing each long drive line with a symmetric difference between itself and the corresponding partial drive line gives us two short drive lines shown in Fig. 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 with another line Q ν It is easy to see that replacing X^ with ^ will generate 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] 8a)-8b) show examples of constraint term placements with partitioning into 3-body and 4-body constraints. The computational problem can be an energy minimization of the general form including k-body interactions for any k, as shown in Eq. 1. Each interaction term in Eq. 1 is mapped to a parity qubit. As shown in Eqs. 8a)-8b), the parity qubits are labeled by tuples of the form 04, 135, etc., where parity qubit 04 corresponds to a 2-body interaction between two classical spins s0 and s4 under the parity mapping, parity qubit 135 corresponds to a 3-body interaction between three classical spins s1, s3, and s5, etc. FIG. 8a) shows a special case with only 4-body constraint terms, where the driving lines are chosen to be strictly horizontal or vertical, and can be trivially parallelized. FIG. 8b) shows a general case including both 3-body and 4-body constraint terms. Explicitly implementing all 3-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 third constraint is H^ C(hatched triangles). The drive lines for the top row of parity qubits are omitted since they can be obtained by symmetric difference with 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 straight lines, as seen in Fig. 8a). This also applies to a mixed 3-body and 4-body layout where all 3-body constraints are explicitly enforced (see Fig. 8b). A possible strategy to split 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 drive. The drive circuit can be run in two steps, with all horizontal drive lines and all vertical drive lines running in parallel, respectively.

[0145] Further optimizations can further reduce the number of explicitly enforced constraint terms. Some of the 3-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 3-regime constraint terms. An example of such an optimization is shown in Figure 9.

[0146] Figure 9 shows an example of an optimized set of explicitly enforced constraint terms (hatched shapes) such that a hybrid driving line that preserves the remaining constraint terms (unhatched) can be implemented in a parallelizable quantum circuit of reasonable depth. The unhatched triangles indicate 3-regime constraint terms that can be implicitly implemented without significantly increasing the required circuit depth. The hatched squares are 4-regime constraint terms that remain explicitly enforced to connect adjacent explicitly enforced 3-regime constraint terms to the 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 circuitry) can scale 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 circuitry 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 modularization of a larger layout 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 constraint terms (hatched boxes and triangles) define the boundaries of each module. Because the hybrid drive line does not need to traverse rows or columns of explicitly enforced constraint terms, each module can be treated individually when building 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., there are strictly vertical and horizontal lines only), the length of the hybrid drive lines must be at most l max become.

[0152] The quantum circuits implementing 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-determined quantity and 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 within a module are implicitly enforced, the problem of finding a suitable hybrid driveline is reduced to a small, separate problem for each module, and the length of the hybrid driveline is still approximately l max It is.

[0154] Thus, in Fig. 10, all solid lines (running horizontally) and all dashed lines (running vertically) can be executed in parallel, respectively. The dotted lines (which may have both horizontal and vertical parts) arise from the implicitly enforced 3-constraint, but have a small contribution to the depth and can be partially parallelized with other steps. In each submodule, the "missing" driving lines are omitted because they can be obtained by symmetric differencing of the other submodules.

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

[0156] As an initial quantum state for the optimization procedure, the considered 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 spanning (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 with 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^ that 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 quantum state is a stabilizer state, so known methods for preparing stabilizer states can be used to build a quantum circuit that generates the 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, let |D|=log2dim(H hyb ) driving qubit (satisfying all the implicitly enforced constraints), which can be seen to (μ) This can be done by defining one qubit per driving line such that acts as the bit-flip operator for that driving qubit. In this diagram, the desired initial state corresponds to all driving qubits being in the |+> state.

[0158] The 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 need to satisfy the following (anti)commutation relations for μ ≠ ν:

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[0159] The goal is to make all the driving qubits μ X^ (μ) -eigenstate |+>. This is easy to prepare, and satisfies the constraints for all driving qubits being in the |↑> state. [×]K (Space H parity To prepare the desired state, we start with the quantum system H parity This is a series of rotations e -iπ / 4X ^ (μ) and e -iπ / 4Z ^ (μ) can be decomposed into

[0160] However, problems arise when a qubit is involved in multiple driving lines. z Performing an operation affects all involved drive lines and can result in unwanted crosstalk that needs to 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 the qubits with index 0 each participate in only one drive line, as shown in Figure 7(b).

[0161] If this is not possible, the drive line Q μ Z^ for qubit k in ∋k (μ) The operation is performed on the other driving line Q that contains qubit k. ν All the driving qubits associated with Z^ (ν)As long as it is in the eigenstate of and is not affected by rotation, it can be executed. [×]K In this case, all the driving qubits are in the Z^ eigenstate. This allows us to (μ) For each rotation, a sequence of drive rotations can be found such that there is at least one qubit on the corresponding drive line that is not included in any other drive line, or that participates only in other drive lines whose states have 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] FIG. 11 shows the quantum circuit depth (vertical axis) required to execute a single step of the QAOA protocol for a layout as shown in FIG. 7. The horizontal axis represents the relative amount of explicit constraint terms. Said relative amount can be modified by considering a different partition of the constraint terms into explicit and implicit execution (i.e., a different partition of the set of summand constraint Hamiltonians into a first and a second subset, as described herein). The left side corresponds to a fully implicit execution. As the amount of explicitly enforced constraint terms increases, the enforcement of the three constraint terms is made explicit one by one at first, and finally all three constraint terms are explicitly enforced at the point marked by the cross. Then, an additional lattice of explicitly enforced constraint terms is introduced for modularization, and the lattice spacing is reduced stepwise.

[0164] The relative amount of constraints that are explicitly enforced, n C / n C totStarting from =0, the circuit depth increases linearly with system size. The large factor in the scaling of the circuit depth is due to the excessive overlap of driving lines in the fully implicit setup (Figure 7b), which makes parallel execution impossible. The 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. The circuit depth scales linearly with system size, but with a much smaller a priori factor. Further improvements are possible through layout modularization. This increases the circuit depth a small amount by a certain number due to the additional execution cost of the constraint Hamiltonian. Since all explicitly enforced constraints in a modularized grid can be executed in parallel, the increase in depth is independent of the system size. However, it is now possible to further reduce the circuit depth required for the driving terms. If the grid is large enough, the relationship between the reachable circuit depth and the relative amount of explicitly enforced constraints becomes independent of the 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 p=3 QAOA cycles, 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 since the constraint part has been removed. At each initialization, we find a local optimum using the following classical procedure: We perform a random QAOA parameter update. If the energy of the system decreases after a parameter update, the new parameters are updated with probability p accept = 1, 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 held for each step. After optimization, the residual energy E of the system state as a function of the number of explicitly enforced constraints is 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 take into account 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 preserves 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 realized on any grid arrangement 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 with 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, i.e., the qubits used for the rotation can be chosen freely. This freedom can be exploited to minimize the circuit depth of a sequence of such operators. Appendix B Partition constraints and driving line search

[0170] In the following we outline a general algorithm for finding valid hybrid driving lines (hereafter simply called lines). Depending on how we choose the constraints / qubits / driving lines (but in principle any choice works), certain steps can be improved (solutions are found faster, the solution has fewer explicitly enforced constraints, or the solution has a smaller circuit depth). Furthermore, the line constraints can follow different criteria, but the algorithmic approach remains the same. As explained in section 3.3, we start with all constraints being implicitly enforced, apart from a grid of explicitly enforced constraints for modularity. We look at 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 (not yet reached) implicitly enforced 4-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., they do not commute with the corresponding driving term), choose one of these constraints. Then extend the line by a qubit that is on the same plaquette (2 × 2 square) as the chosen constraint and adjacent to an existing line qubit, but not yet part of the line. If no such qubit exists, proceed to the "failure module". -Repeat the above steps (select another unenforced 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 the overlapping lines), remove the longest line until the remaining lines are independent (i.e. 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, the maximum length of the line, the branching / bending of the line (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: · "Failed 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 failed 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) and ensure that all explicitly enforced constraints are connected to the boundaries (through adjacent explicitly enforced constraints). Note that multiple constraints may be needed to connect to the boundaries, and there can 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 when 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 generic hybrid drivelines, but are only necessary for the true hybrid case, since 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 you to assign priorities to all lines, update the explicit / implicit grouping of constraints according to Appendix B and try again. At the latest, it works 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 lines that are not in the driving Hamiltonian. 0-3: Priority P of lines that perform rotations. μ (P μ =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, which is 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 sum of summand constraint Hamiltonians (252), a ground state of the sum Hamiltonian encoding a solution to the computational problem, the sum Hamiltonian comprising a sum of the problem Hamiltonian and the constraint Hamiltonian; A first subset of the summand constraint Hamiltonians of the constraint Hamiltonian (S １ ), and a second subset of the summand constraint Hamiltonians of the constraint Hamiltonian (S ２ ), and performing N operations (N≧2), each operation comprising: A procedure for preparing an initial quantum state; 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, 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 summation of augend problem Hamiltonians of the problem Hamiltonian, each constraint-enforcing unitary operator being 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 a unitary time evolution operator of a summation of augend constraint Hamiltonians taken from the first subset, and each unitary driving operator being 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 of the operations comprises executing at least some of the unitary operators of the sequence by a quantum circuit comprising quantum gates.

3. 3. The method of claim 1 or 2, 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. A method of performing quantum computing as recited in claim 3, wherein each unitary driving operator operates entirely within one of the subsystems of the quantum system.

5. 5. The method of 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 of performing quantum computing as claimed in claim 3, wherein each unitary driving operator is realised by a quantum circuit of constant depth.

7. 7. A method for performing quantum computing as claimed in 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 method for performing quantum computing according to claim 1 , wherein the driving Hamiltonian commutes with all augend constraint Hamiltonians of the second subset of the augend 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 sum of two or more augend driving Hamiltonians of the driving Hamiltonian.

9. The sequence of at least a portion of unitary operators among the N operations is A, where p≧3. １ A ２ ...A ｐ or at least a subsequence of said form, 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-encoded unitary operator, and X ｉ , Y ｉ and Z ｉ Another one of them is a constraint-enforcing unitary operator, X ｉ , Y ｉ and Z ｉ 9. A method for performing quantum computing as claimed in claim 1, wherein yet another one of said operators is a unitary driving operator.

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

11. 11. The method of performing quantum computing of 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 performing the measurement in the first operation includes measuring an 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 in a second time includes measuring an 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 a sequence of unitary operators to be applied in a third one of the N operations, the third one being performed after the second one, the sequence of unitary operators to be applied in the third one 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 Each C^ has the form ｌ is C^ ｌ = a ｌ Z^ ｌ +b ｌ I, and Z^ ｌ is Pauli σ ｚ is the tensor product of operators, I is the identity operator, and a ｌ and b ｌ are coefficients, and each C^ ｌ 13. A method for performing quantum computing as claimed in claim 1, wherein: t 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 the 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 method of 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 sum of summand constraint Hamiltonians (252), a ground state of the sum Hamiltonian encoding a solution to the computational problem, the sum Hamiltonian comprising a sum of the problem Hamiltonian and the constraint Hamiltonian; A first subset of the summand constraint Hamiltonians of the constraint Hamiltonian (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 being configured to perform N (N≧2) operations, each of which comprises: 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, each constraint-enforcing unitary operator the evolving step, wherein an operator 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 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). A device that performs quantum computing.

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