Efficient Fault-Tolerant Trotter Simulation of the Molecular Hamiltonian

Abstract

Methods, systems, and apparatuses for determining properties of a physical system described by an electronic structure Hamiltonian. In one aspect, a Hamiltonian describing a physical system is transformed into a qubit Hamiltonian describing a corresponding qubit system. The qubit Hamiltonian includes a plurality of two-qubit interaction terms, each term including a corresponding translation-invariant coefficient. The qubit system evolves under a unitary operator generated by the plurality of two-qubit interaction terms. The evolution includes applying a multi-layer quantum logic gate to the qubit system, where each application of a layer evolves the qubit system under a unitary operator generated by a corresponding subset of the plurality of two-qubit interaction terms, and where the values of the coefficients of the subset of the plurality of two-qubit interaction terms generating the unitary operator are constant. The evolved qubit system is measured, and a property of the physical system is determined.

Description

Technical Field

[0001] This specification relates to quantum computing. Background Art

[0002] A quantum simulator is a device designed to provide information and insights about a physical system or device. Quantum simulators enable physical systems that are difficult to study in the laboratory or cannot be modeled using classical processors to be simulated. Summary of the Invention

[0003] This specification describes techniques for performing fault-tolerant Trotter simulation of a molecular Hamiltonian.

[0004] Generally, an innovative aspect of the subject matter described in this specification can be implemented in a method for determining properties of a physical system, the method comprising: transforming a Hamiltonian describing the physical system into a qubit Hamiltonian describing a corresponding qubit system, wherein the qubit Hamiltonian includes a plurality of two-qubit interaction terms, each two-qubit interaction term including a corresponding translation-invariant two-qubit interaction term coefficient; evolving the qubit system under a unitary operator generated by the plurality of two-qubit interaction terms, including: applying a multi-layer quantum logic gate to the qubit system, wherein each application of a layer in the multi-layer evolves the qubit system under a unitary operator generated by a corresponding subset of the plurality of two-qubit interaction terms; wherein the values of the two-qubit interaction term coefficients of the subset of the plurality of two-qubit interaction terms generating the unitary operator are constant; measuring the evolved qubit system; and determining one or more properties of the physical system based on the measurement of the evolved qubit system.

[0005] Other embodiments of this aspect include corresponding computer systems, devices, and computer programs recorded on one or more computer storage devices, each of the foregoing being configured to perform the actions of the method. A system of one or more classical and / or quantum computers can be configured to perform specific operations or actions by virtue of software, firmware, hardware, or a combination thereof installed on the system, the software, firmware, hardware, or a combination thereof causing the system to perform the actions when operating. One or more computer programs can be configured to perform specific operations or actions by virtue of instructions that, when executed by a data processing device, cause the device to perform the actions.

[0006] The foregoing and other embodiments may optionally include, individually or in combination, one or more of the following features. In some embodiments, each layer of the multi-layer quantum logic gate includes a rotation operation having the same rotation angle.

[0007] In some embodiments, the method further includes generating multi-layer quantum logic gates. Generating multi-layer quantum logic gates includes, for a system of an even number N of qubits: arranging the N qubits as a two-dimensional grid of qubits, where the position of each qubit in the grid is represented by a position vector p; for each index vector s: defining a set of qubit pairs (p, p + s) for each position vector p over all N qubits; based on the defined set of qubit pairs, generating two layers of qubit interactions, where each qubit in each layer interacts with another qubit.

[0008] In some embodiments, each layer in the multi-layer quantum logic gates includes N / 2 equiangular rotation operations.

[0009] In some embodiments, the method further includes generating multi-layer quantum logic gates. Generating multi-layer quantum logic gates includes, for a system of an odd number N of qubits: arranging the N qubits as a two-dimensional grid of qubits, where the position of each qubit in the grid is represented by a position vector p; for each index vector s: defining a set of qubit pairs (p, p + s) for each position vector p over all N qubits, where one qubit is not included in the defined set of qubit pairs; generating three layers of qubit interactions based on the defined set of qubit pairs, including: a first layer where each qubit interacts with another qubit, a second layer where each qubit interacts with another qubit, and a third layer including a single rotation operation.

[0010] In some embodiments, both the first layer and the second layer of the quantum logic gates include (N - 1) / 2 equiangular rotation operations.

[0011] In some embodiments, evolving the qubit system under the unitary operator generated by a plurality of two-qubit interaction terms further includes: in each layer of the multi-layer, applying Hamming weight phasing to the rotations in that layer.

[0012] In some embodiments, defining a set of qubit pairs (p, p + s) for each qubit p includes evaluating p + s modulo the grid length.

[0013] In some embodiments, the qubit Hamiltonian further includes a kinetic energy term and a single-qubit potential energy term, and wherein the method further includes evolving the qubit system under the unitary operator generated by the kinetic energy term and the unitary operator generated by the single-qubit potential energy term.

[0014] In some embodiments, evolving a qubit system under a unitary operator generated by multiple two-qubit interaction terms, kinetic energy terms, or single-qubit potential energy terms includes implementing time evolution or implementing directed control of evolution on an ancilla qubit.

[0015] In some embodiments, evolving a qubit system under a unitary operator generated by multiple two-qubit interaction terms, kinetic energy terms, or single-qubit potential energy terms includes evolving the qubit system in a split-operator Trotter step.

[0016] In some embodiments, the split-operator Trotter step is a step that produces the minimum Trotter error among multiple split-operator Trotter steps.

[0017] In some embodiments, the Hamiltonian describing the physical system includes an electronic structure Hamiltonian.

[0018] In some embodiments, the physical system includes a chemical substance or material, and wherein determining the properties of the physical system based on measurements of the evolved qubit system includes using the simulated evolution of the qubit system under the qubit Hamiltonian to determine the properties of the chemical substance or material.

[0019] The subject matter described in this specification can be implemented in a particular manner so as to achieve one or more of the following advantages.

[0020] The presently described disclosure represents a significant and widely applicable improvement over the prior art in quantum simulation.

[0021] When using a Trotter-based method to determine (e.g., simulate) the evolution of an electronic structure Hamiltonian, the evolution of the interaction part of the potential operator typically appears to require applying O(N 2 ) rotations to N qubits. The techniques described in this application reduce the number of required applications of rotations to O(N log N) at the cost of 4(N - 1)(N - 2) additional T gates. Since the number of T gates required to synthesize each rotation is much greater than 4, this significantly reduces the fault-tolerant cost of determining these terms.

[0022] Additionally, the techniques described in this specification can be applied to any quantum simulation of an electronic structure Hamiltonian using a plane-wave dual basis, or more generally to any simulation of two-qubit interaction terms with the translational invariance property of equation (3) below. Thus, a variety of different settings can benefit from the techniques described. For example, the techniques can be used to more efficiently design and test materials or catalysts, more accurately predict the rates of certain chemical reactions, or more efficiently synthesize drugs.

[0023] Details of one or more embodiments of the subject matter of this specification are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter will become apparent from the description, the drawings, and the claims. BRIEF DESCRIPTION OF THE DRAWINGS

[0024] Figure 1 Depicts an example system for determining properties of a physical system.

[0025] Figure 2 Is a flowchart of an example process for determining properties of a physical system.

[0026] Figure 3A Is a flowchart of an example process for determining interactions in each layer of a multi-layer quantum logic gate applied to an even number of qubits.

[0027] Figure 3B Is a flowchart of an example process for determining interactions in each layer of a multi-layer quantum logic gate applied to an odd number of qubits.

[0028] Figure 4 Is a diagram of two example layers of qubit interactions.

[0029] Like reference numerals and designations in the various drawings indicate like elements. DETAILED DESCRIPTION

[0030] The electronic structure Hamiltonian describes the properties of electrons interacting in the presence of fixed nuclei. The physics of these systems determines the rates of chemical reactions and the properties of most materials. Thus, determining the electronic structure Hamiltonian has applications in a variety of different fields ranging from drug synthesis to the design of new catalysts and materials.

[0031] The dynamic properties of these systems are typically very difficult to solve for classical computers. Thus, a variety of methods for quantum simulation of electronic structure have been developed, including second quantization, first quantization, algorithms in real space; adiabatic algorithms; methods based on the Taylor series method of time-evolution and other algorithms using linear combinations of unitary operator frameworks; variational algorithms; Bravyi-Kitaev encoding; reduced locality Jordan-Wigner encoding and low-rank tensor decomposition of Coulomb operators. However, the only method that has been consistently compiled into a universal gate set and has been rigorously evaluated for feasibility within error correction is Trotter-based methods in a second-quantized basis.

[0032] This specification describes new and improved techniques built on such Trotter-based methods. This technique reduces the number of rotation operations required to implement the evolution under the interaction part of the potential energy operator in the split-operator Trotter step. Generally, the evolution under the interaction part of the potential energy operator requires O(N 2 ) rotations for N qubits. However, by working in the plane wave dual basis and constructing an ordered layer of quantum logic gates where equiangular rotations are grouped together, Hamming weight phasing can be applied, and the evolution cost under the interaction part of the potential energy operator can be reduced to O(N log N) rotations.

[0033] Example Hardware

[0034] Figure 1 Example system 100 is depicted for determining a physical system described by an electronic structure Hamiltonian. Example system 100 is an example of a system implemented as a classical or quantum computer program on one or more classical computers or quantum computing devices at one or more locations, in which the systems, components, and techniques described below can be implemented.

[0035] System 100 may include quantum hardware 102 that communicates data with classical processor 104. System 100 may receive data as input, which may include data representing a physical system of interest, e.g., input data 106. The received data representing the physical system of interest may include data representing a Hamiltonian that describes the physical system to be modeled or simulated. In some implementations, the received data may include data representing an electronic structure Hamiltonian that describes the corresponding physical system, e.g., a single molecule, material, or chemical substance.

[0036] This system may produce data representing the results of determining the physical system of interest as output, e.g., output data 108 representing the determined physical properties of the physical system. For example, as described above, in some implementations, the physical system may be a chemical substance. In these cases, the data representing the simulation results can be used to determine the properties of the chemical substance, e.g., the rate of a chemical reaction. As another example, also as described above, in some implementations, the physical system may be a material, e.g., a semiconductor. In these cases, the data representing the simulation results can be used to determine the properties of the material, e.g., conductivity.

[0037] System 100 is configured to perform classical computing combined with quantum computing using classical processor 104 and quantum hardware 102. Classical processor 104 may include a Hamiltonian transformation module 114 that is configured to process received input data representing a Hamiltonian that describes a physical system, e.g., input data 106, and produce data representing a qubit Hamiltonian that describes a qubit system corresponding to the physical system. For example, Hamiltonian transformation module 114 may be configured to apply a Jordan-Wigner transformation to received input data 106 to produce data representing the corresponding qubit Hamiltonian.

[0038] Quantum hardware 102 is configured to determine the evolution of the physical system specified by input data 106, e.g., by simulating using corresponding data representing a qubit Hamiltonian that describes the qubit system. For example, as described in detail below with reference to Figures 2-4 Quantum hardware 102 may be configured to implement a Trotter step, e.g., a split-operator Trotter step 118, by applying a quantum circuit of unitary operators that affect the state of the system of qubits 110 included in the quantum hardware and that depend on the qubit Hamiltonian.

[0039] The type of qubits 110 included in quantum hardware 102 depends on various factors, e.g., the type of physical system to be determined or simulated or laboratory constraints, and may vary. For example, in some cases, it may be convenient to include one or more resonators attached to one or more superconducting qubits (e.g., Gmon or Xmon qubits). In other cases, ion traps, photonic devices, or superconducting cavities (with which states can be prepared without qubits) may be used. Other example implementations of qubits include fluxmon qubits, silicon quantum dots, or phosphorus-doped qubits.

[0040] Qubits 110 may be arranged in a grid with controllable interactions between the qubits. One or more control devices 112 include devices such as arbitrary waveform generators that are configured to apply a quantum circuit of quantum logic gates (e.g., rotation operations) to qubits 110, e.g., by applying corresponding control pulses along control lines that couple qubits 110 to control devices 112. Control devices 112 may also include devices configured to perform measurements on qubits 110 and provide measurement results 120 to classical processor 104, e.g., readout resonators.

[0041] As described in detail in step 208 below with reference to Figure 2 the measurement results 120 received by classical processor 104 may be processed to determine properties of the physical system specified by input data 106.

[0042] Programming the Hardware

[0043] Figure 2 is a flowchart of an example process 200 for determining properties of a physical system. For convenience, process 200 will be described as being performed by a system of one or more classical or quantum computing devices located at one or more locations. For example, a quantum computing system appropriately programmed according to this specification, such as Figure 1 system 100, can perform process 200.

[0044] The system transforms the Hamiltonian describing the physical system into a qubit Hamiltonian describing the corresponding qubit system (step 202). For example, the Hamiltonian describing the physical system can be a Fermi Hamiltonian of the following form:

[0045]

[0046] where T pq 、U p and V pq represent numbers defined by integrals over molecular orbitals, caused by kinetic energy, external potential, and electron-electron Coulomb repulsion, respectively, and the creation and annihilation operators ap, are indexed by vectors p and q such as lattice sites and / or spin orbitals. T pq 、U p and V pq can represent real numbers. In some embodiments, one or more of T pq and / or V pq can be complex numbers as long as the overall Hermitian property of the Hamiltonian is maintained. In some embodiments, the Hamiltonian H can be a Hamiltonian describing a material (e.g., a polymer aircraft wing or rocket, a solar cell, a battery, a catalytic converter, or a thin-film electronic device), or a Hamiltonian describing a chemical substance.

[0047] The Hamiltonian in the form of Equation (1) can be transformed into a qubit Hamiltonian describing the corresponding qubit system using the Jordan-Wigner transformation. The Jordan-Wigner transformation transforms the Fermi annihilation and creation operators and a p into the Pauli operators X p 、Y p 、Z p corresponding to spin-1 / 2 particles. The standard form of the Jordan-Wigner transformation is given by known Given. Transforming the Hamiltonian of Equation (1) using the Jordan-Wigner transformation yields the qubit Hamiltonian given in Equation (2) below, where the identity term is omitted as it is readily determined.

[0048]

[0049] In Equation (2), X p , Y p and Z p denote the Pauli X, Y, and Z gates, respectively, which act on qubits indexed by p in Jordan-Wigner ordering, denotes a string of Pauli Z gates between qubit p and qubit q in Jordan-Wigner order (necessary to maintain the appropriate commutation relations of the fermionic creation and annihilation operators for the transformation) and and denote coefficients that can be directly calculated from the corresponding coefficients in the second-quantized Hamiltonian given in Equation (1) above.

[0050] The Hamiltonian of Equation (1) can be expressed in one of several different bases. For convenience, in Equation (1), the Hamiltonian H is written in the plane wave basis. However, one or more terms of the Hamiltonian H can also be expressed in the plane wave dual basis. The plane wave dual basis can be obtained by applying the unitary discrete Fourier transform to the plane wave basis.

[0051] In the plane wave dual basis, the qubit Hamiltonian H qubit includes multiple two-qubit interaction terms with translation-invariant coefficients That is, for any vector s of a particular qubit in the indexed qubit system:

[0052]

[0053] To determine the properties of a physical system, the system performs a quantum simulation of the evolution of the physical system under the Hamiltonian H using a qubit system and a qubit Hamiltonian. Performing the quantum simulation includes determining the evolution of the physical system under the Hamiltonian H by evolving the qubit system under the unitary operator generated by the qubit Hamiltonian, using a variant of the split-operator algorithm.

[0054] The split-operator algorithm can be used to split the evolution under the Hamiltonian H qubit under Split into separate evolutions under the kinetic and potential energy parts of the Hamiltonian. This is particularly useful because the terms in the kinetic energy part of the Hamiltonian (those with coefficient ) can be diagonalized by an efficient circuit transformation C, resulting in a sum of single-qubit terms, namely:

[0055]

[0056] The diagonalizing circuit C, Changing between the position and momentum bases using the fast Fourier transform for fermions or Givens rotations. In the split-operator algorithm, the Trotter-Suzuki approximation can be applied to divide the evolution under the total Hamiltonian H qubit into two evolutions: the first evolution e p under the kinetic energy term T = C(∑ p T p ) and the second evolution e -iTt under the potential energy term and -i(U+V)t .

[0057] Symmetrize these evolutions to give a second-order Trotter step that approximates the time evolution under the total Hamiltonian by evolution blocks separated by the circuit C and its inverse Equation (4) describes a single second-order Trotter step. The approximation can be refined by dividing t into such steps (each with time t / r).

[0058]

[0059]

[0060] There are multiple choices for how to symmetrize the split-operator Trotter step. Instead of using equation (4), which approximates the evolution of r Trotter steps as e -iHt ≈ (e -iTt / 2r e -i(U+V)t / r e -iTt / 2r e ) r , an alternative choice approximates the evolution of r Trotter steps as e -iHt ≈ (e -i(U+V)t / 2r e -iTt / r e -i(U+V)t / 2r ) r , where the single-order Trotter step in this form is given by equation (5) below.

[0061]

[0062] When different circuits are implemented in the split-operator steps given by equation (4) or (5), each term or A rotation is required that can be arbitrarily rotated, whether for implementing time evolution or for the evolution that can be directed controlled on the ancilla used for phase estimation. Therefore, using N spin-orbitals, at most N(N + 3) / 2 rotations are required to determine the evolution under all terms in the Hamiltonian, and each Trotter step circuit can be completed by applying the basis change circuit C and its inverse once.

[0063] At first glance, the number of gates required for the two Trotter steps given by equations (4) and (5) may seem different. The number of terms in T is linear in the number of spin-orbitals, while the number of terms in U + V is quadratic in the number of spin-orbitals. Therefore, it can be expected that equation (5) in which e appears twice requires more gates than equation (4) in which e appears only once. On the other hand, C or appears four times in equation (4), but only twice in equation (5), which may indicate that equation (4) can use more gates. -i(U+V)t -i(U+V)t However, this intuition is deceptive. Since the number of Trotter steps r is usually very large, for example, r > 400, and the start and end of the two split-operator steps are merged together, the cost difference is negligible. The result of this merging is that the cost of the r-Trotter step simulation is almost the same for the two competing orderings of the terms. In the last Trotter step, the two orderings are different: compared with equation (5), equation (4) applies C one more time and e one less time. Each of the remaining r - 1 Trotter steps in both variants applies C, e, t, and e once, completely outweighing this final difference.

[0064] Instead, the difference in the number of Trotter steps r required for accurate simulation (determined by the Trotter error of the two steps) determines the cost difference. Therefore, when performing the example process 200, the selected Trotter step construction depends on which split-operator step produces a smaller Trotter error. -i(U+V)t -iTt -i(U+V)t

[0065]

[0066] When implementing the evolution of a qubit system under the unitary operators given by equations (4) and (5) above, the system implements the evolution of the unitary operator generated by the two-qubit interaction terms in a layer, such that the determination (e.g., simulation) within a given layer ​​​​​​​​​​Each value of Figure 4 is the same (step 204). That is, the system applies a multi-layer quantum logic gate to the qubit system, where each application of a layer in the multi-layer causes the qubit system to evolve under a unitary operation generated by a corresponding subset of multiple two-qubit interaction terms, and the values of the two-qubit interaction term coefficients for the subsets of two-qubit interaction terms that generate the unitary operator are constant. The method for selecting the interactions to be simulated in each layer of the multi-layer quantum logic gate is described in more detail below with reference to FIGS. 3 and

[0067] By sorting the simulation of the terms in in this way, all the rotation operations used in each layer have the same rotation angle. This enables the rotation operations within each layer to be fully combined using Hamming weight phasing. Hamming weight phasing uses M-1 auxiliary qubits to reduce M parallel rotations of the same angle to rotations of different angles, along with 4M-4 additional T gates. Since synthesizing an arbitrary rotation is very costly on a fault-tolerant quantum computer, this can result in significant cost savings. An example application of Hamming weight phasing is described below.

[0068] The system measures the evolved qubit system (step 206). In some embodiments, this can include measuring the phase accumulated during the evolution as determined on a separate auxiliary qubit. For example, this can allow the determination of the electronic ground state energy for a single value of a set of nuclear coordinates.

[0069] Based on the measurement of the evolved qubit system, the system determines one or more properties of the physical system (step 208).

[0070] For example, in some embodiments, the Hamiltonian describing the physical system can characterize a chemical substance. In these embodiments, the system can perform steps 204 and 206 for individual nuclear coordinates (depending on the specific chemical substance characterized by the Hamiltonian), where measuring the evolved qubit system includes measuring the energy as a function of the individual nuclear coordinates. Such measurement results can be used to generate a potential energy surface from which the rate of a chemical reaction can be determined.

[0071] As another example, in some embodiments, a Hamiltonian that describes a physical system can characterize a material. In these embodiments, the system can measure the electronic ground state energy and use the ground state energy to determine the physical properties of the material. For example, information about the equilibrium crystal structure of the material can be determined by performing measurements on a plurality of evolving qubit systems, where each system represents a different set of nuclear coordinates of the material. Based on these measurements, the ground state energy can be determined for each set of nuclear coordinates. The lowest ground state energy determined from these measurements can correspond to the crystal structure of the material. Other ground state energies can be used to determine other physical properties of the system, such as phase transitions between different structures. Other properties that can be determined for the material by simulating the Hamiltonian in this way include: magnetic order; electronic structure; electrical properties of the material, such as conductivity; and / or mechanical properties, such as Young's modulus and / or compressibility.

[0072] Figure 3A is a flowchart of an example process 300 for determining the interactions in each layer of a multi-layer quantum logic gate applied to an even number of qubits. For convenience, process 300 will be described as being performed by a system of one or more classical or quantum computing devices located at one or more locations. For example, a quantum computing system appropriately programmed according to this specification, such as Figure 1 system 100, can perform process 300.

[0073] The system arranges N qubits as a two-dimensional grid of qubits, where the position of each qubit in the grid is represented by a position vector p (step 302). Figure 4 An illustration of an example 4×4 grid of 16 qubits is shown in. The position vector is given by a two-dimensional vector. In some embodiments, the components of the vector are integers that label the positions in the two-dimensional grid of qubits.

[0074] For each index vector s, the system defines a set of qubit pairs {(p, p + s)} for each position vector p over all qubits (step 304). For all qubit position vectors p, the coefficient translational invariance ensures that:

[0075]

[0076] When the number N of qubits is even, there are N such qubit pairs (p, p + s), which are given by N - 1 different index vectors s on the grid. However, since each of these interactions is between two qubits, at most N / 2 interactions can be performed simultaneously in one layer. Therefore, for each index vector s, the system generates two layers of qubit interactions based on a defined set of qubit pairs, where each qubit in each layer interacts with another qubit (step 306). In each layer, N / 2 interactions of equal strength can be simulated, resulting in N / 2 equiangular rotations that can be combined using Hamming weight phasing. In Figure 4 Two example layers of qubits for the interaction with index vector s = (0, 1) are shown. The index vector is given by a two-dimensional vector. In some embodiments, the components of the vector are integers that label the position difference between two qubits in a two-dimensional grid of qubits. For example, s = (m, n), where m and n are integers that label the position difference between two qubits along two perpendicular axes.

[0077] Generally, there are N - 1 different index vectors s that must be iterated because the final vector s = 0 does not occur because it corresponds to a qubit interacting with itself. Each index vector s is applied in two layers, resulting in 2N - 2 layers, with each layer determining N / 2 interactions. In each of the 2N - 2 layers, Hamming weight phasing can be applied to each term because each coefficient is the same, and thus each of the N / 2 rotation angles is the same. This allows Hamming weight phasing to be applied to each rotation in each layer, reducing the number of rotations required per layer from N / 2 to Over all 2N - 2 layers, the T cost for all two-qubit terms in the steps for determining the Trotter evolution is where T synth = O(log(1 / ∈)) represents the number of T gates required to synthesize each rotation, where ∈ represents the required synthesis precision.

[0078] Figure 3B is a flowchart of an example process 350 for determining the interactions in each layer of a multi-layer quantum logic gate applied to an odd number of qubits. For convenience, process 350 will be described as being executed by a system of one or more classical or quantum computing devices located at one or more locations. For example, a quantum computing system appropriately programmed according to this specification, such as Figure 1 system 100, can execute process 350.

[0079] The system arranges N qubits as a two-dimensional grid of qubits, where the position of each qubit in the grid is represented by a position vector p (step 352).

[0080] For each index vector s, the system defines a set of qubit pairs {(p, p + s)} for each position vector p over all qubits (step 354). Since the number of qubits is odd, one qubit will not be included in the defined set of qubit pairs. Similarly, for all qubit position vectors p, the translational invariance of the coefficient guarantees

[0081] When the number of qubits N is odd, three layers are used instead of two to simulate each of the N - 1 index vectors s. This is because, when the number of qubits is odd, one qubit is left over in each layer when defining qubit pairs. As a result, there are (N - 1) / 2 layers of 2N - 2 equally angled rotations and N - 1 layers of individually rotated qubits. Then, Hamming weight phasing can be applied to reduce the (N - 1) / 2 equally angled rotations in each layer to Over all layers, the T cost for all two-qubit terms in the steps for determining Trotter evolution is

[0082] For both even and odd N, at most 4(N - 1) 2 extra T gates are used to reduce the cost of rotation to O(N log N log(1 / ∈)). This is to be compared with the O(N 2 log(1 / ∈)) cost when Hamming weight phasing is not applied or the interaction terms are not translationally invariant. This improvement applies to any quantum simulation of electronic structure using plane-wave dual bases, or more generally, to any simulation of two-qubit interactions with the translational invariance property given by equation (3).

[0083] Figure 4Figure 400 of two example layers (a) and (b) of qubit interactions. For the case of 16 qubits on a 4×4 grid, two example layers are generated for the index vector s = (0, 1). In these two layers, qubit p interacts with qubit p + s, where the elements of the index vectors p and p + s are evaluated modulo the grid length. For example, considering the first row 410 of the 4×4 grid in layer (a), the qubit 402 at position p = (0, 0) interacts with the qubit 404 at position p + s = (0, 0)+(0, 1) = (0, 1), and the qubit 406 at position p = (0, 2) interacts with the qubit 408 at position p + s = (0, 2)+(0, 1) = (0, 3). In layer (b), the qubit 404 at position p = (0, 1) interacts with the qubit 406 at position p + s = (0, 1)+(0, 1) = (0, 2), and the qubit 408 at position p = (0, 3) interacts with the qubit 402 at position p + s = (0, 3)+(0, 1) = (0, 4) ≡ (0, 0) mod(4).

[0084] Since each interaction (indicated by a dashed line, e.g., line 412) is between two qubits, at most [N = 16] / 2 = 8 interactions can be performed simultaneously in the layer. Thus, for the index vector s = (0, 1), two layers (a) and (b) with 8 qubit interactions are generated, where each qubit in each layer interacts with another qubit.

[0085] Example Application of Hamming Weight Phase Locking

[0086] Consider a circuit that applies the same single - qubit rotation R z (θ)=exp(−iθZ / 2) simultaneously to three different qubits. The action on the logical states is different phases that depend only on the Hamming weight of the logical state. Specifically,

[0087] 1. The all - zero state |000> takes the phase - 3θ / 2,

[0088] 2. The three states |001>, |010>, |100> with Hamming weight 1 each take the phase - θ / 2,

[0089] 3. The three states |011>, |101>, |110> with Hamming weight 2 are all phase - shifted by θ / 2, and

[0090] 4. The all - one state |111> takes the phase 3θ / 2.

[0091] The Hamming weight of the input state can be calculated without applying the same angle R zThree rotations of (θ), and two different rotations can be applied to the Hamming weight: applying R z (θ) to the 1s bit, and applying R z (2θ) to the 2s bit. In this case,

[0092] 1. Apply the phase -3θ / 2 to the all - zero state |000> (Hamming weight 0),

[0093] 2. Apply -θ / 2=-θ + θ / 2 to |001>, |010>, |100> (in binary, Hamming weight 1 = 01 b ),

[0094] 3. Apply θ / 2=+θ - θ / 2 to |011>, |101>, |110> (Hamming weight 2 = 10 b ), and

[0095] 4. Apply

[0096] For both processes, the phase on each logical state is the same. However, since costly T - gates must be used to synthesize the rotations, reducing the number of rotation gates in the circuit reduces its fault - tolerance cost.

[0097] This idea can be easily generalized to the case of n repeated equiangular rotations R z (θ) occurring in parallel in the circuit: Instead of applying n original rotations, the Hamming weight of the relevant qubits can be calculated, and instead, rotations R z (θ), R z (2θ), R z (4θ),... can be applied to the Hamming weight. This technique is called Hamming - weight phasing.

[0098] The digital and / or quantum topics described in this specification, as well as the implementations of digital functional operations and quantum operations, can be implemented in digital electronic circuits, suitable quantum circuits (or more generally, quantum computing systems), in tangibly embodied digital and / or quantum computer software or firmware, in digital and / or quantum computer hardware (including the structures disclosed in this specification and their structural equivalents), or in a combination of one or more of them. The term "quantum computing system" can include, but is not limited to, quantum computers, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0099] The embodiments of the digital and / or quantum subject matter described in this specification can be implemented as one or more digital and / or quantum computer programs, i.e., one or more modules of digital and / or quantum computer program instructions encoded on a tangible non-transitory storage medium for execution by, or to control the operation of, a data processing apparatus. The digital and / or quantum computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits, or a combination of one or more of them. Alternatively or additionally, the program instructions can be encoded on an artificially generated propagated signal (e.g., a machine-generated electrical, optical, or electromagnetic signal) capable of encoding digital and / or quantum information, which is generated to encode digital and / or quantum information for transmission to a suitable receiver device for execution by the data processing apparatus).

[0100] The terms "quantum information" and "quantum data" refer to information or data carried, held, or stored in a quantum system, where the smallest non-trivial system is a qubit, i.e., the system that defines the unit of quantum information. It should be understood that the term "qubit" encompasses all quantum systems that can be appropriately approximated as two-level systems in the corresponding context. Such quantum systems can include multi-level systems, e.g., having two or more levels. By way of example, such systems can include atoms, electrons, photons, ions, or superconducting qubits. In many embodiments, the computational basis states are identified with the ground state and the first excited state, but it should be understood that other settings identifying the computational states with higher excited states are also possible.

[0101] The term "data processing apparatus" refers to digital and / or quantum data processing hardware and encompasses all kinds of devices, equipment, and machines for processing digital and / or quantum data, including, by way of example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, multi-digital and quantum processors or computers and their combinations. The apparatus can also be or further include special-purpose logic circuitry (e.g., FPGA (field-programmable gate array), ASIC (application-specific integrated circuit)) or a quantum simulator (i.e., a quantum data processing apparatus designed to simulate or generate information about a specific quantum system). In particular, a quantum simulator is a special-purpose quantum computer that does not have the ability to perform universal quantum computing. In addition to the hardware, the apparatus can optionally include code that creates an execution environment for the digital and / or quantum computer program, e.g., code constituting processor firmware, protocol stack, database management system, operating system, or a combination of one or more of them).

[0102] A digital computer program (which may also be referred to or described as a program, software, software application, module, software module, script, or code) can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program (which may also be referred to or described as a program, software, software application, module, software module, script, or code) can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and can be converted into a suitable quantum programming language or can be written in a quantum programming language (such as QCL or Quipper).

[0103] A digital and / or quantum computer program may or may not correspond to a file in a file system. The program can be stored in a part of a file that holds other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple collaborating files (e.g., files that store some parts of one or more modules, subroutines, or code). A digital and / or quantum computer program can be deployed to execute on one digital computer or one quantum computer or on multiple digital and / or quantum computers, which are located at one site or distributed across multiple sites and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that can use quantum systems (such as qubits) to transmit quantum data. Generally speaking, a digital data communication network cannot transmit quantum data, but a quantum data communication network can transmit both quantum data and digital data.

[0104] The processes and logical flows described in this specification can be performed by one or more programmable digital and / or quantum computers, which operate, as appropriate, with one or more digital and / or quantum processors, executing one or more digital and / or quantum computer programs to perform functions by operating on input digital and quantum data and generating output. The processes and logical flows can also be performed by, or the apparatus can also be implemented as, special-purpose logic circuitry (e.g., FPGA or ASIC) or a quantum simulator, or by a combination of special-purpose logic circuitry or a quantum simulator and one or more programmed digital and / or quantum computers.

[0105] The meaning that a system of one or more digital and / or quantum computers “is configured to” perform a specific operation or action is that software, firmware, hardware, or a combination thereof is installed on the system, and when operating, the software, firmware, hardware, or a combination thereof causes the system to perform the operation or action. The meaning that one or more digital and / or quantum computer programs are configured to perform a specific operation or action is that one or more programs include instructions that, when executed by a digital and / or quantum data processing device, cause the device to perform the operation or action. A quantum computer may receive instructions from a digital computer, and the instructions, when executed by the quantum computing device, cause the device to perform the operation or action.

[0106] A digital and / or quantum computer suitable for executing a digital and / or quantum computer program may be based on a general-purpose or special-purpose digital and / or quantum processor or both, or any other kind of central digital and / or quantum processing unit. Generally speaking, the central digital and / or quantum processing unit will receive instructions and digital and / or quantum data from a read-only memory, a random access memory, or a quantum system (such as a photon) suitable for transmitting quantum data, or a combination thereof.

[0107] The basic elements of a digital and / or quantum computer are a central processing unit for executing or running instructions and one or more memory devices for storing instructions and digital and / or quantum data. The central processing unit and the memory may be supplemented by dedicated logic circuits or quantum simulators or may be incorporated therein. Generally speaking, a digital and / or quantum computer will also include one or more mass storage devices (such as magnetic, magneto-optical, optical disks) for storing digital and / or quantum data or a quantum system suitable for storing quantum information, or is operatively coupled to receive digital and / or quantum data from it or transmit digital and / or quantum data to it or receive and transmit both. However, a digital and / or quantum computer does not have to have such devices.

[0108] Digital and / or quantum computer-readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include all forms of non-volatile digital and / or quantum memories, media, and memory devices. For example, they include semiconductor memory devices such as EPROM, EEPROM, and flash memory devices; magnetic disks such as internal hard disks or removable disks; magneto-optical disks; CD-ROM and DVD-ROM disks; and quantum systems such as trapped atoms or electrons. It should be understood that a quantum memory is a device that can store quantum data for a long time with high fidelity and high efficiency, such as a light-matter interface, where light is used for transmission and matter is used for storing and maintaining the quantum characteristics of quantum data, such as superposition or quantum coherence.

[0109] Control of the various systems or portions of systems described in this specification can be implemented in a digital and / or quantum computer program product that includes instructions stored on one or more non-transitory machine-readable storage media and executable on one or more digital and / or quantum processing devices. The systems or portions of systems described in this specification can each be implemented as an apparatus, a method, or a system that can include one or more digital and / or quantum processing devices and a memory for storing executable instructions to perform the operations described in this specification.

[0110] Although this specification contains many specific implementation details, these details should not be construed as limiting the scope that can be claimed, but rather as descriptions of features that can be specific to particular implementations. Certain features described in the context of separate implementations in this specification can also be implemented in combination in a single implementation. Conversely, the various features described in the context of a single implementation can also be implemented separately in multiple implementations or in any suitable sub-combination. Additionally, although features may be described as acting in certain combinations and even initially claimed as such, in some cases, one or more features from the claimed combination can be excluded from the combination, and the claimed combination can be directed to a sub-combination or a variant of the sub-combination.

[0111] Similarly, although operations are depicted in the figures in a particular order, this should not be construed as requiring that the operations be performed in the particular order shown or in a sequential order, or that all illustrated operations be performed to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous. Additionally, the separation of the various system modules and components in the above-described implementations should not be construed as requiring such separation in all implementations, but rather it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products.

[0112] Particular implementations of the subject matter have been described. Other implementations are within the scope of the appended claims. For example, the acts recited in the claims can be performed in a different order and still achieve the desired result. As an example, the processes depicted in the figures do not necessarily need the particular order shown or a sequential order to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. A method for determining properties of a physical system, the method being performed by a system comprising one or more classical processors and quantum hardware, the quantum hardware comprising: A system of N interacting qubits arranged on a two-dimensional grid, wherein the interactions between the qubits are controllable, and wherein the position of each qubit in the grid is represented by a position vector p; and a plurality of control devices configured to apply quantum logic gates to the system of interacting qubits and measure the system of interacting qubits, the method comprising: Transforming, by the one or more classical processors, the Hamiltonian describing the physical system into a qubit Hamiltonian describing a corresponding qubit system, wherein the qubit Hamiltonian comprises a plurality of two-qubit interaction terms, each two-qubit interaction term comprising a corresponding translation-invariant two-qubit interaction term coefficient; Evolving, by the quantum hardware, the qubit system under a unitary operator generated by the plurality of two-qubit interaction terms, comprising: Generating a multi-layer quantum logic gate, comprising for each index vector s: When N is even, defining a set of qubit pairs (p, p + s) for each position vector p over all N qubits, or when N is odd, defining a set of qubit pairs (p, p + s) for each position vector p over all qubits except one qubit among all N qubits; and Based on the defined set of qubit pairs, generating two layers of qubit interactions, wherein each qubit in each layer interacts with another qubit; Applying the multi-layer quantum logic gate to the qubit system, wherein each application of a layer in the multi-layer evolves the qubit system under a corresponding unitary operator generated by a corresponding subset of the plurality of two-qubit interaction terms, and wherein each layer in the multi-layer quantum logic gate comprises a rotation operation with the same rotation angle, and wherein evolving the qubit system under the unitary operator generated by the plurality of two-qubit interaction terms further comprises: in each layer of the multi-layer, applying Hamming weight phasing to the rotation in that layer, wherein applying Hamming weight phasing comprises: Determining the Hamming weight for the qubits in that layer; and Applying the rotation to the qubits based on the Hamming weight; wherein the values of the two-qubit interaction term coefficients of the subset of the plurality of two-qubit interaction terms generating the corresponding unitary operator are constant; Measuring, by the quantum hardware, the evolved qubit system; and Based on the measurement of the evolved qubit system, determining, by the one or more classical processors, one or more properties of the physical system.

2. The method according to claim 1, wherein, N is even and wherein each layer in the multi-layer quantum logic gate comprises N / 2 equiangular rotation operations.

3. The method according to claim 1, wherein, N is odd and wherein generating the multi-layer quantum logic gate further comprises: Generate another layer of qubit interactions based on a defined set of qubit pairs, such that the layer of qubit interactions includes: A first layer, wherein each qubit interacts with another qubit, A second layer, wherein each qubit interacts with another qubit, and A third layer, the third layer including a single rotation operation.

4. The method according to claim 3, wherein, Both the first layer and the second layer of the quantum logic gates include (N - 1) / 2 equiangular rotation operations.

5. The method according to claim 1, wherein, Defining a set of qubit pairs (p, p + s) for each qubit p includes evaluating p + s modulo the grid length.

6. The method according to claim 1, wherein, The qubit Hamiltonian further includes a kinetic energy term and a single - qubit potential energy term, and wherein the method further includes evolving the qubit system under the unitary operator generated by the kinetic energy term and the unitary operator generated by the single - qubit potential energy term.

7. The method according to claim 6, wherein, Evolving the qubit system under the unitary operator generated by the multiple two - qubit interaction terms, the kinetic energy term, or the single - qubit potential energy term includes implementing time evolution or implementing a directed control - led evolution on an ancilla qubit.

8. The method according to claim 6 or 7, wherein, Evolving the qubit system under the unitary operator generated by the multiple two - qubit interaction terms, the kinetic energy term, or the single - qubit potential energy term includes evolving the qubit system in a split - operator Trotter step.

9. The method according to claim 8, wherein, The split - operator Trotter step is the one that produces the minimum Trotter error among multiple split - operator Trotter steps.

10. The method according to claim 1, wherein, The Hamiltonian describing the physical system includes an electronic - structure Hamiltonian.

11. The method according to claim 1, wherein, The physical system includes a chemical substance or a material, and wherein determining the properties of the physical system based on the measurement of the evolved qubit system includes using the simulated evolution of the qubit system under the qubit Hamiltonian to determine the properties of the chemical substance or the material.

12. An apparatus for determining the properties of a physical system, the apparatus comprises: One or more classical processors; Quantum hardware, including: A system of interacting qubits, wherein the interaction between qubits is controllable; and Multiple control devices configured to apply quantum logic gates to the system of interacting qubits and measure the system of interacting qubits; wherein the apparatus is configured to perform the method according to any one of claims 1 - 11.

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