Quantum circuits for unitary evolution of low-energy states

Abstract

Methods, systems, and apparatus for Hamiltonian simulation of low-energy initial quantum states. In one aspect, a quantum circuit that performs unitary time evolution for a Hamiltonian is determined. The Hamiltonian is mapped to a corresponding gap-amplifiable Hamiltonian with a low energy subspace, the gap-amplifiable Hamiltonian comprising one or more matrices, where each matrix comprises a respective block encoding that is dependent on a parameter λ. An intermediate Hamiltonian is determined, where eigenvalues of the intermediate Hamiltonian are the same as eigenvalues of a square root of the gap-amplifiable Hamiltonian. A block encoding of the intermediate Hamiltonian divided by the square root of λ is determined. A filter is applied to the block encoding to obtain a polynomial approximation of the unitary time evolution operator for the Hamiltonian. The quantum circuit is applied to an initial state that is supported in the low-energy subspace.

Description

[0001]QUANTUM CIRCUITS FOR UNITARY EVOLUTION OF LOW-ENERGY STATES CROSS-REFERENCE TO RELATED APPLICATION This application claims the benefit of U.S. Patent Application No.63 / 623,152 filed on January 19, 2024, which is incorporated by reference herein in its entirety. BACKGROUND One example application of quantum computing is simulating the dynamics of quantum systems. Given a Hamiltonian, an initial quantum state, and an evolution time, the task is to produce an evolved quantum state as determined by Schrödinger’s equation. Simulating the dynamics of a quantum system, also known as Hamiltonian simulation, is relevant to numerous problems in physics, chemistry, and is used in many quantum algorithms. Low-energy quantum states of a Hamiltonian are states that are supported only on a subspace of energies that are strictly below the largest energy (eigenvalue) of the Hamiltonian. Hamiltonian simulation of low-energy states is relevant in physically motivated contexts, such as when studying zero-temperature quantum phase transitions in condensed matter systems or simulating quantum field theories, and in quantum chemistry, such as when computing ground-state energies of molecules using quantum phase estimation. It is also important for adiabatic quantum computing, a technique to solve optimization and other problems in quantum computing by preparing the ground state of a complex (interacting) Hamiltonian starting from the ground state of a simpler (non-interacting) Hamiltonian. SUMMARY This disclosure relates to quantum computing. In particular, this disclosure describes techniques for generating quantum circuits for unitary evolution of low-energy quantum states. Innovative aspects of the subject matter described in this specification can be implemented in methods that include: determining a quantum circuit that performs unitary time evolution for a first Hamiltonian, comprising: mapping the first Hamiltonian to a corresponding gap-amplifiable Hamiltonian with an energy subspace specified by a parameter Δ, the gap-amplifiable Hamiltonian comprising one or more matrices, wherein each matrix comprises a respective block encoding that is dependent on a parameter ^^; computing a block encoding of an intermediate Hamiltonian divided by a square root of ^^, wherein eigenvalues of the intermediate Hamiltonian are the same as eigenvalues of a square root of the gap-amplifiable Hamiltonian; and applying a filter to the block encoding of the intermediate Hamiltonian divided by the square root of ^^ to obtain a polynomial approximation of the unitary time evolution operator for the first Hamiltonian; and applying the quantum circuit to an initial state that is supported in the energy subspace of the gap- amplifiable Hamiltonian. Other implementations of these aspects includes corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods. A system of one or more classical and quantum computers can be configured to perform particular operations or actions by virtue of having software, firmware, hardware, or a combination thereof installed on the system that in operation causes or cause the system to perform the actions. One or more computer programs can be configured to perform particular operations or actions by virtue of including instructions that, when executed by data processing apparatus, cause the apparatus to perform the actions. The foregoing and other implementations can each optionally include one or more of the following features, alone or in combination. In some implementations the gap- amplifiable Hamiltonian is a Hamiltonian ^^ with ground state energy ^^^and low energysubspace ^^^^,^^^ ^ Δ^ such that ^^ െ ^^ ൌ ∑^ற^ୀ^ ^^^^^^ , where ^^ ^ ^^^, ^^^ െ ^^ ൌ ^^^Δ^, and^^^^^^^ୀ^represents the one or In some implementations the method further comprises obtaining the block encodings of the one or more matrices; or constructing the block encodings of the one or more matrices using an oracle model. In some implementations the oracle model comprises a linear combination of unitaries model or a sparse model. In some implementations the quantum circuit comprises ^^ ^ ^ఒ^ఢ^ queries to the block encoding of the intermediate Hamiltonian divided ^^, wherein each query is included in the quantum circuit and comprises one or more quantum logic gates. In some implementations each query comprises a query to a controlled block-encoding unitary ^^, wherein the block-encoding unitary satisfies Π^^Π ൌ ୌି^^ where Π represents a projector operator, ^^ represents the gap- ^^, and ^^ represents an energy that is less than or equal to the ground of the gap- amplifiable Hamiltonian. In some implementations the polynomial approximation is obtained over an interval ^െ^^ఒ ,^^ఒ^, wherein t represents time and^ఒ^ 1.In some implementations computing the block encoding of the intermediate divided by the square root of λ comprises performing spectral gap amplification. In some implementations applying the filter to the block encoding of the intermediate Hamiltonian divided by the square root of ^^ to obtain the polynomial approximation of the unitary time evolution operator for the Hamiltonian comprises performing a quantum singular value transform. In some implementations applying the quantum circuit to the initial state comprises: combining sine and cosine components of the polynomial approximation; applying a linear combination of unitaries transformation to the combined sine and cosine components to obtain a block encoding of the unitary time evolution operator for the first Hamiltonian; and compiling the quantum circuit according to the block encoding of the unitary time evolution operator for the first Hamiltonian. The subject matter described in this specification can be implemented in particular ways so as to realize one or more of the following advantages. Hamiltonian simulation of low-energy states is relevant in physically motivated contexts, such as when identifying zero-temperature quantum phase transitions in condensed matter systems, simulating quantum field theories, or computing ground-state energies via quantum phase estimation as in quantum chemistry. It is also important for adiabatic quantum computing, a generic technique to solve problems in optimization by preparing the ground state of a complex Hamiltonian starting from the ground state of a simpler one. When utilizing the presently-described techniques to further understand specific physical systems, inputs to the process may include one or more measurements which characterize physical properties of the physical system, and / or information obtained through simulation may be used to control or calibrate the physical system. For example, techniques may include constructing and / or selecting the first Hamiltonian to reflect one or more measurements characterizing the physical system. Known techniques for Hamiltonian simulation (different to the presently described techniques) include methods based on Trotter-Suzuki product formulas when the Hamiltonian is presented as a sum of positive semidefinite terms or product formula algorithms such as qDRIFT or randomized product formulas. Such techniques are not optimal in terms of the evolution time or the allowed error. Other known techniques include using quantum phase estimation in combination with spectral gap amplification. However, these techniques are also not optimal because of the considerable computational overhead of quantum phase estimation with the allowed error. Other known techniques are based on uniform spectral amplification, however such techniques are not known to be applicable to low energy states. The presently described techniques improve on the above described known techniques. For example, the presently described techniques can be used to prepare a quantum circuit that implements unitary time evolution of a low energy state (an initial state confined to the subspace corresponding to eigenvalues [-1,-1 + Δ / λ] for Δ ≤ λ) with bounded error (^) with optimal query complexity. In particular, the quantum circuit includes only^^ ^^^√^^Δ ^ ^ఒ ^log ^queries to a unitary block encoding of a shifted version of the t represents evolution time, Δ, λ defined the low energy subspace of the initial state, and ^ represents a target error threshold. This represents a logarithmic improvement to, e.g., techniques that are based on uniform spectral amplification whichrequire a larger number of qubits, e.g., ^^ ^^^√^^Δ logଷ / ଶ௧ఒ^ ^ఒହ / ଶ ௧ఒ^ log ఢ ^ queries. The reduction in the required number of of the shifted version of the Hamiltonian fast forwards time evolution. The presently described techniques therefore represent a significant improvement compared to existing general approaches to Hamiltonian simulation, which exhibit a linear dependence of the query complexity on λ, as well as approaches tailored specifically for low-energy states. Since each query is physically realized as a sequence of quantum logic gates, optimizing (i.e. reducing) the number of queries included in the quantum circuit optimizes (i.e., reduces) the circuit depth. This in turn, optimizes (i.e. reduces) the number of operations and amount of time required by a quantum computer to execute the quantum circuit and may reduce the number of components of the quantum computer (such as qubits and / or control circuitry) necessary to execute the circuit. In addition, errors in the quantum circuit / quantum computer are reduced, e.g., because computation time per qubit is reduced. Details of one or more implementations 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 FIG.1 is a block diagram of an example system for Hamiltonian simulation of low- energy initial quantum states. FIG.2 is a flow diagram of an example process for performing unitary time evolution of a quantum system that is initialized in a low energy quantum state of a corresponding Hamiltonian. FIG.3 shows an example quantum computing device. DETAILED DESCRIPTION This disclosure describes techniques for Hamiltonian simulation of low-energy initial quantum states. A quantum circuit for implementing the time evolution operator is constructed using techniques such as spectral gap amplification and the quantum singular value transform. The quantum circuit is applied to a register of qubits that encode a low energy initial quantum state to simulate the time evolution of the low-energy initial quantum state. FIG.1 is a block diagram of an example system 100 for Hamiltonian simulation of low-energy initial quantum states. The system 100 is an example of a system implemented as classical and quantum computer programs on one or more classical computers and quantum computing devices in one or more locations, in which the systems, components, and techniques described herein can be implemented. The example system 100 includes a quantum processor operating system 102 and a quantum processor 104. Components of the example system can be connected through a network, e.g., a local area network (LAN), wide area network (WAN), the Internet, or a combination thereof. The quantum processor operating system 102 is configured to perform classical computations and operations that control and manage the quantum processor 104. The quantum processor operating system 102 can be implemented as one or more computer programs, e.g., one or more modules of 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 computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, or a combination of one or more of them. The quantum processor 104 includes physical components for performing quantum computations. For example, the quantum processor 104 can include registers of physical qubits 106 and control devices 108 configured to operate the physical qubits 106 and apply quantum circuits to the qubits. In some implementations the quantum processor 102 can be a noisy device, e.g., a noisy intermediate-scale quantum device (NISQ device), a trapped ion quantum computer, or a neutral atom quantum computer. The example system 100 is configured to receive as input a request for Hamiltonian simulation of low-energy initial quantum state 110. The low-energy initial quantum state is defined by a parameter Δ. The request 110 is provided to a quantum circuit compiler 112 included in the quantum processor operating system and may include one or more measurements characterizing a physical system embodying the quantum state. The quantum circuit compiler 112 is configured to construct a quantum circuit that performs unitary time evolution for the Hamiltonian. To construct the quantum circuit, the quantum circuit compiler 112 maps the Hamiltonian to a corresponding gap-amplifiable Hamiltonian with an energy subspace specified by the parameter Δ. The gap-amplifiable Hamiltonian is dependent on a parameter ^^. The quantum circuit compiler 112 then determines a spectral gap amplification Hamiltonian with eigenvalues that are the same as a square root of the gap-amplifiable Hamiltonian. The quantum circuit compiler 112 computes a block encoding of the spectral gap amplification Hamiltonian and applies a filter to the block encoding of the spectral gap amplification Hamiltonian divided by the square root of λ to obtain a polynomial approximation of the unitary time evolution operator for the first Hamiltonian. The quantum circuit compiler 112 is configured to determine unitary operators that represent the polynomial approximation, e.g., through application of a linear combination of unitaries transformation, and decompose the unitary operators into native qubit gates for the quantum processor 104. Example operations performed by the quantum circuit compiler 112 are described in more detail below with reference to FIG.2. The quantum processor operating system 102 includes control software 114 that is configured to cause the quantum processor 104 to implement the quantum circuit of native qubit gates 116. Results of the implementation of the quantum circuit, e.g., measured results, are provided to a post processor 118 included in the quantum processor operating system 102 for post processing, e.g., converting into classical data. The quantum processor operating system 102 is then configured to output classical data representing results of the Hamiltonian simulation 12 FIG.2 is a flow diagram of an example process 200 for performing unitary time evolution of a quantum system that is initialized in a low energy quantum state of a corresponding Hamiltonian. For convenience, the process 200 will be described as being performed by a quantum processor operating system. For example, the quantum processor operating system 102 of FIG.1, appropriately programmed in accordance with this specification, can perform the process 200. To simulate the Hamiltonian ^^ for a time ^^ ^ 0 and with a predetermined errortolerance ^^ ^ 0, the system constructs a quantum circuit that performs unitary time evolution(step 202). That is, the system constructs a quantum circuit that performs the unitaryoperation ^^ି^ு௧with error at most ^^. For a quantum system with Hamiltonian ^^, the state ofthe system during the simulation at time ^^, |^^^^^^^, satisfies |^^^^^^^ ൌ ^^ି^ு௧|^^^0^^ where|^^^0^^ represents an initial quantum state, e.g., a low energy quantum state. To construct the quantum circuit, the system maps the Hamiltonian ^^ to a corresponding gap-amplifiable Hamiltonian with a low energy subspace specified by a parameter Δ (step 204). The gap-amplifiable Hamiltonian can be written as a single term^^^^ற^^ or a linear combination of terms ^^ ൌ ∑^ ^ୀ^ ^^^^^ற^^^^. In some implementations each term in the combination can represent a local interaction between spins or other quantum systems. A general gap-amplifiable Hamiltonian can be defined as follows. Consider a Hamiltonian acting on a Hilbert space. Assume that a quantum state in the Hilbert space issupported exclusively in a low energy subspace ^^^^ corresponding to eigenvalues ^^^^,^^^ ^ Δ^൧of the Hamiltonian for some Δ^ ^ 0, where ^^^ represents the ground state of the Hamiltonian.Let ^^ ∈ ℝ and ^^^ ^ 0 for all ^^ ∈ ^^^^. The Hamiltonian is then said to be ^Δ, ^^^ െgap-amplifiable if efficient access to block encodings of operators ^^^ is provided such that ^^ െ^^^^ே ൌ ∑^ ^ୀ^ ^^^^^ற^^^^and the state ∣ψ^ is supported exclusively in the subspace correspondingto energies at most Δ ^ Δ^ of ^^ െ ^^^^ே.That is, the gap-amplifiable Hamiltonian includes (i.e., is defined by) one or more matrices, where each matrix has a respective block encoding that is dependent on a parameter ^^. In particular, the gap-amplifiable Hamiltonian is a Hamiltonian ^^ with ground stateenergy ^^ and low energ ^ ^ ∑^ற^y subspace ^^^,^^^ ^ Δ such that ^^ െ ^^ ൌ ^ୀ^ ^^^^^^ , where ^^ ^^^^, ^^^ െ ^^ ൌ ^^^Δ^, and ^^^^^^^ୀ^represents the one or more matrices. The matrices ^^^can berepresented as ^^ ൈ ^^ matrices. In some implementations, M is not equal to N. In theseimplementations the matrices ^^^can be padded with zeros so that the matrix is square. This can simplify the construction of the block encoding of the matrixes since spaces of different dimension are avoided. The block encodings of the matrices ^^^can be efficiently accessed. In this context, efficient access refers to direct access to each block-encoding of ^^^and ^^ற^ or a constant number of queries to an oracle. A block encoding of a matrix can be directly accessed if the matrix can be implemented (applied to a quantum state) directly, e.g., if the block encoding is a unitary operator that can be directly included in the quantum circuit. A block encoding of a matrix can be accessed via queries to an oracle model if an oracle is included in the quantum circuit in order to apply the block encoding selectively. An oracle is an operation that applies a unitary operator based on a given input. The oracle can be implemented in the quantum circuit as a subroutine or set of quantum gates that apply the block encoding depending on an input state of a control register. For example, the system can construct the block encodings of the matrices included in the gap amplifiable Hamiltonian using a known oracle model such as the linear combination of unitaries (LCU) model or sparse model. In the LCU model, a constant number of queries to SELECT and PREPARE oracleswith ^^ ൌ ∑ ^^^^^^^^^^gives a ^^^, ^^^-gap-a| ^ ^|ଶ^^ ^ ^ mplifiable Hamiltonian with ^^ ൌ ^^m^ax^^ ^^^. The is an operation that applies a conditional operation based on the state of a control register. The select oracle selects and applies specific unitary operators from a setof operators. For example, the SELECT oracle can implement a unitary operator ^^^^^^^௧ suchthat ^^^^^^^௧|^^^|^^^ ൌ |^^^^^^^|^^^, where ∣j^ is a control register specifying the index j, ∣ψ^ is aquantum state that the selected unitary ^^^acts upon, and ^^^is a j-th unitary operator from a predefined set of operators. The PREPARE oracle is an operation that prepares a superposition state that encodes coefficients or weights associated with a decomposition. For example, the PREPARE oracle can implement a unitary operator ^^^^^^^^^such that ^^^^^^^^^|0^ ൌ ∑^ ^^^|^^^ , where |0^ is an initial state of the control register, ^^^ are coefficients,e.g., from a Hamiltonian decomposition, and |^^^is a computational basis state for the index j. prepare oracles can be used to implement a target operation, where the prepare oracle prepares a superposition of indices that encode coefficients in a decomposition of the target operation and the select oracle is conditionally executes operations in the decomposition of the target operation based on the superposition state. As another example, in the sparse model, a constant number of queries to ^^ுand ^^ிoracles with d-sparse ^^^ gives a ^^^, ^^^-gap-amplifiable Hamiltonian with ^^ ൌ^^^ଶ ^^^^ଶmaxห|^|ห^^௫. The ^^ுoracle is an operation that implements a matrix as a block encoding using a combination of control registers that guide application of the elements of the matrix. The ^^ிoracle is an operation that prepares a superposition of quantum states that represent different elements of a matrix and conditionally applies specific components of the matrix based on the quantum state. Together, the ^^ுand ^^ிoracles can be used to implement a target operation, where the ^^ிoracle prepares a quantum state so that the state contains information about the target operation, e.g., eigenvalues, and the ^^ுoracle applies a block encoding of the target operation to the quantum state, allowing the quantum state to evolve according to the target operation without needing to store or manipulate the entire target operation. The system then determines an intermediate Hamiltonian ^^ௌீ^(also referred to herein as a spectral gap amplification Hamiltonian) divided by a square root of ^^ (step 206) and computes a block encoding of an intermediate Hamiltonian ^^ௌீ^divided by a square root of ^^, e.g., using spectral gap amplification (step 208). In the case that the gap- amplifiable Hamiltonian is written as ^^^^ற^^, the intermediate Hamiltonian ^^ௌீ^can be determined as a matrix, where block diagonal components of the matrix are zero, a lower left off-diagonal block of the matrix is equal to A and the upper right off-diagonal block is equal to ^^ற, where the matrix is multiplied by a factor of √^^. That is, By construction, the ^^ଶௌீ^ ൌ ^^^ 00 ^^^^ற^^^. That is, the first block of ^^ௌଶீ^contains the Hamiltonian H to be Hamiltonian can be considered a square root of the Hamiltonian can be diagonalized to show that its eigenvalues are the square roots of the eigenvalues of ^^ଶ. Therefore, it is possible to simulate transformations on H, such as the time evolution operator ^^ି^௧ு, using the intermediate Hamiltonian ^^ௌீ^instead. For example, it can be shown that మ ห0^൫^^ି^௧ுห^^^൯ ൌ ^^ି^௧^ுೄಸಲ^మ ห0^|^^^ ൌ ^^ି^௧ఒ^ಹೄಸಲ√ഊ^ |0^|^^^. (2) ^మ for any space. can be implemented using the quantum singular value transform in cases that access to a block encoding of the intermediate Hamiltonian ^^ௌீ^divided by the square root of ^^ is given, as discussed below. The system applies a filter to the block encoding of the intermediate Hamiltonian divided by the square root of ^^, e.g., using a quantum singular value transform (QSVT), to obtain a polynomial approximation of the unitary time evolution operator for the Hamiltonian (step 210). In a setting where a block-encoding of ^^ / ^^ ൌ ^^ற^^ is provided, application of theQSVT applies even and odd filters to the eigenvalue ^^ ∈ ^െ||^^|| / ^^, ||^^|| / ^^^. That is, x ispassed through filters cos^^^^^^^^ and sin^^^^^^^^. These are then combined with a linear combination of unitaries (a technique in quantum computing where multiple unitary operators are combined in a linear superposition) to obtain a block encoding of eି^௧ு ൌ cos^^^^^^^^ െ^^ sin^^^^^^^^. Given the block encoding of the intermediate Hamiltonian divided by the squareroot of ^^, the spectrum is instead supported on ^^ ∈ ^െ^||^^|| / ^^,^||^^|| / ^^^ since ห|^^ௌீ^|ห ൌ^||^^||. Therefore, the system obtains polynomial approximations to cos^^^^^^^ଶ^ sin^^^^^^^ଶ^ that will allow for the simulation of cos^^^^^^ and sin^^^^^^ using the block encoding of the intermediate Hamiltonian divided by the square root of ^^. The system replaces ^^ → ^^ௌீ^ / √^^ in the polynomials to implement the target quantum operationమ ି^௧ ಹೄಸಲ^^ఒ^^ , which simulates the time evolution operator ^^ . Since the initial state for the supported only on the subspace corresponding to eigenvalues of the Hamiltonian to be simulated that are at most Δ, the polynomial approximation is obtained over a domain interval ^െ^^,^ ^ఒ ^ ఒ^, where ఒ^ 1.The time evolution operator can be used to compile the quantum singular value transform can be performed on the polynomial approximation of the unitary time evolution operator for the Hamiltonian to combine sine and cosine components of the polynomial approximation. A linear combination of unitaries (LCU) transformation can then be applied to the combined sine and cosine components to obtain a block encoding of the unitary time evolution operator for the Hamiltonian. The quantum circuit can then be compiled according to the block encoding of the unitary time evolution operator for the Hamiltonian, e.g., as a sequence of controlled queries to the block encoding. For example, application of the LCU transformation can output a weighted sum of unitary operators that represents the polynomial approximation. The system can then decompose the unitary operators into native qubit gates for the quantum computer. In some cases the LCU transformation introduces an ancilla register prepared a quantum state that encodes the weights of the weighted sum. Coupling the ancilla register to the target register (which holds the quantum state that is being evolved) enables controlled application of the unitary operators in the weighted sum. The system applies the quantum circuit to a register of qubits that are prepared in an initial state that encodes of a low energy initial state, e.g., an initial state that is supported in the energy subspace ^0,Δ^ of the block encoding of the gap-amplifiable Hamiltonian (step212). By construction, the quantum circuit comprises ^^ ^^^√^^Δ ^ ^ఒ^^logఢ^ queries to the block encoding of the intermediate Hamiltonian of ^^, where each query is included in the quantum circuit and consists of one or more quantum logic gates. Each query is a query to a controlled block-encoding unitary ^^ (the block encoding of the intermediate Hamiltonian divided by the square root of ^^) where the block-encoding unitarysatisfies Π^^Π ൌ ୌି^^ where Π represents a projector operator, ^^ represents the gap- amplifiable ^^, and ^^ represents an energy that is less than or equal to the ground state energy ^^^of the gap-amplifiable Hamiltonian. The system outputs a result of the application of the quantum circuit to the low energy initial state (step 214). For example, the system can measure the register of qubits after the quantum circuit has been applied to obtain a measured result of the simulation. FIG.3 depicts an example quantum computer 300 for performing the quantum operations described in this specification. The example quantum computer 300 includes an example quantum computing device 302. The quantum computing device 302 is intended to represent various forms of quantum computing devices. The components shown here, their connections and relationships, and their functions, are exemplary only, and do not limit implementations of the inventions described and / or claimed in this document. The example quantum computing device 302 includes a qubit assembly 352 and a control and measurement system 304. The qubit assembly includes multiple qubits, e.g., qubit 306, that are used to perform algorithmic operations or quantum computations. While the qubits shown in FIG.3 are arranged in a rectangular array, this is a schematic depiction and is not intended to be limiting. The qubit assembly 352 also includes adjustable coupling elements, e.g., coupler 308, that allow for interactions between coupled qubits. In the schematic depiction of FIG.3, each qubit is adjustably coupled to each of its four adjacent qubits by means of respective coupling elements. However, this is an example arrangement of qubits and couplers and other arrangements are possible, including arrangements that are non-rectangular, arrangements that allow for coupling between non-adjacent qubits, and arrangements that include adjustable coupling between more than two qubits. Each qubit can be a physical two-level quantum system or device having levels representing logical values of 0 and 1. The specific physical realization of the multiple qubits and how they interact with one another is dependent on a variety of factors including the type of the quantum computing device 302 included in the example computer 300 or the type of quantum computations that the quantum computing device is performing. For example, in an atomic quantum computer the qubits may be realized via atomic, molecular or solid-state quantum systems, e.g., hyperfine atomic states. As another example, in a superconducting quantum computer the qubits may be realized via superconducting qubits or semi-conducting qubits, e.g., superconducting transmon states. As another example, in a NMR quantum computer the qubits may be realized via nuclear spin states. As another example, in a neutral atom quantum computer the qubits may be realized via an array of neural atoms, e.g., rubidium or cesium, where Rydberg interactions allow for implementations of multi-qubit gates and facilitate connectivity between qubits in the array. In some implementations a quantum computation can proceed by loading qubits, e.g., from a quantum memory, and applying a sequence of unitary operators to the qubits. Applying a unitary operator to the qubits can include applying a corresponding sequence of quantum logic gates to the qubits, e.g., to implement the surface code circuits described in this specification. Example quantum logic gates include single-qubit gates, e.g., Pauli-X, Pauli-Y, Pauli-Z (also referred to as X, Y, Z), Hadamard gates, S gates, rotations, two-qubit gates, e.g., controlled-X, controlled-Y, controlled-Z (also referred to as CX, CY, CZ), controlled NOT gates (also referred to as CNOT) controlled swap gates (also referred to as CSWAP), iSWAP gates, and gates involving three or more qubits, e.g., Toffoli gates. The quantum logic gates can be implemented by applying control signals 310 generated by the control and measurement system 304 to the qubits and to the couplers. For example, in some implementations the qubits in the qubit assembly 352 can be frequency tunable. In these examples, each qubit can have associated operating frequencies that can be adjusted through application of voltage pulses via one or more drive-lines coupled to the qubit. Example operating frequencies include qubit idling frequencies, qubit interaction frequencies, and qubit readout frequencies. Different frequencies correspond to different operations that the qubit can perform. For example, setting the operating frequency to a corresponding idling frequency may put the qubit into a state where it does not strongly interact with other qubits, and where it may be used to perform single-qubit gates. As another example, in cases where qubits interact via couplers with fixed coupling, qubits can be configured to interact with one another by setting their respective operating frequencies at some gate-dependent frequency detuning from their common interaction frequency. In other cases, e.g., when the qubits interact via tunable couplers, qubits can be configured to interact with one another by setting the parameters of their respective couplers to enable interactions between the qubits and then by setting the qubit’s respective operating frequencies at some gate-dependent frequency detuning from their common interaction frequency. Such interactions may be performed in order to perform multi-qubit gates. The type of control signals 310 used depends on the physical realizations of the qubits. For example, the control signals may include RF or microwave pulses in an NMR or superconducting quantum computer system, or optical pulses in an atomic quantum computer system. A quantum computation can be completed by measuring the states of the qubits, e.g., using a quantum observable such as X or Z, using respective control signals 310. The measurements cause readout signals 312 representing measurement results to be communicated back to the measurement and control system 304. The readout signals 312 may include RF, microwave, or optical signals depending on the physical scheme for the quantum computing device and / or the qubits. For convenience, the control signals 310 and readout signals 312 shown in FIG.3 are depicted as addressing only selected elements of the qubit assembly (i.e. the top and bottom rows), but during operation the control signals 310 and readout signals 312 can address each element in the qubit assembly 352. The control and measurement system 304 is an example of a classical computer system that can be used to perform various operations on the qubit assembly 352, as described above, as well as other classical subroutines or computations. The control and measurement system 304 includes one or more classical processors, e.g., classical processor 314, one or more memories, e.g., memory 316, and one or more I / O units, e.g., I / O unit 318, connected by one or more data buses. The control and measurement system 304 can be programmed to send sequences of control signals 310 to the qubit assembly, e.g. to carry out a selected series of quantum gate operations, and to receive sequences of readout signals 312 from the qubit assembly, e.g. as part of performing measurement operations. The processor 314 is configured to process instructions for execution within the control and measurement system 304. In some implementations, the processor 314 is a single-threaded processor. In other implementations, the processor 314 is a multi-threaded processor. The processor 314 is capable of processing instructions stored in the memory 316. The memory 316 stores information within the control and measurement system 304. In some implementations, the memory 316 includes a computer-readable medium, a volatile memory unit, and / or a non-volatile memory unit. In some cases, the memory 316 can include storage devices capable of providing mass storage for the system 304, e.g. a hard disk device, an optical disk device, a storage device that is shared over a network by multiple computing devices (e.g., a cloud storage device), and / or some other large capacity storage device. The input / output device 318 provides input / output operations for the control and measurement system 304. The input / output device 318 can include D / A converters, A / D converters, and RF / microwave / optical signal generators, transmitters, and receivers, whereby to send control signals 310 to and receive readout signals 312 from the qubit assembly, as appropriate for the physical scheme for the quantum computer. In some implementations, the input / output device 318 can also include one or more network interface devices, e.g., an Ethernet card, a serial communication device, e.g., an RS-232 port, and / or a wireless interface device, e.g., an 802.11 card. In some implementations, the input / output device 318 can include driver devices configured to receive input data and send output data to other external devices, e.g., keyboard, printer and display devices. Although an example control and measurement system 304 has been depicted in FIG. 3, implementations of the subject matter and the functional operations described in this specification can be implemented in other types of digital electronic circuitry, or in computer software, firmware, or hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. Implementations of the subject matter and operations described in this specification can be implemented in digital electronic circuitry, analog electronic circuitry, suitable quantum circuitry or, more generally, quantum computational systems, in tangibly-embodied software or firmware, in computer hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. The term “quantum computational systems” may include, but is not limited to, quantum computers, quantum information processing systems, quantum cryptography systems, or quantum simulators. Implementations of the subject matter described in this specification can be implemented as one or more computer programs, i.e., one or more modules of computer program instructions encoded on a tangible non-transitory storage medium for execution by, or to control the operation of, data processing apparatus. The 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 in addition, the program instructions can be encoded on an artificially- generated propagated signal that is capable of encoding digital and / or quantum information, e.g., a machine-generated electrical, optical, or electromagnetic signal, that is generated to encode digital and / or quantum information for transmission to suitable receiver apparatus for execution by a data processing apparatus. The terms quantum information and quantum data refer to information or data that is carried by, held or stored in quantum systems, where the smallest non-trivial system is a qubit, i.e., a system that defines the unit of quantum information. It is understood that the term “qubit” encompasses all quantum systems that may be suitably approximated as a two- level system in the corresponding context. Such quantum systems may include multi-level systems, e.g., with two or more levels. By way of example, such systems can include atoms, electrons, photons, ions or superconducting qubits. In many implementations the computational basis states are identified with the ground and first excited states, however it is understood that other setups where the computational states are identified with higher level excited states are possible. The term “data processing apparatus” refers to digital and / or quantum data processing hardware and encompasses all kinds of apparatus, devices, and machines for processing digital and / or quantum data, including by way of example a programmable digital processor, a programmable quantum processor, a digital computer, a quantum computer, multiple digital and quantum processors or computers, and combinations thereof. The apparatus can also be, or further include, special purpose logic circuitry, e.g., an FPGA (field programmable gate array), an ASIC (application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing apparatus that is designed to simulate or produce information about a specific quantum system. In particular, a quantum simulator is a special purpose quantum computer that does not have the capability to perform universal quantum computation. The apparatus can optionally include, in addition to hardware, code that creates an execution environment for digital and / or quantum computer programs, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them. A digital computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it 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, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and translated into a suitable quantum programming language, or can be written in a quantum programming language, e.g., QCL or Quipper. A computer program may, but need not, correspond to a file in a file system. A program can be stored in a portion 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 coordinated files, e.g., files that store one or more modules, sub- programs, or portions of code. A computer program can be deployed to be executed on one computer or on multiple computers that 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 may transmit quantum data using quantum systems, e.g. qubits. Generally, a digital data communication network cannot transmit quantum data, however a quantum data communication network may transmit both quantum data and digital data. The processes and logic flows described in this specification can be performed by one or more programmable computers, operating with one or more processors, as appropriate, executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA or an ASIC, or a quantum simulator, or by a combination of special purpose logic circuitry or quantum simulators and one or more programmed digital and / or quantum computers. For a system of one or more computers to be “configured to” perform particular operations or actions means that the system has installed on it software, firmware, hardware, or a combination of them that in operation cause the system to perform the operations or actions. For one or more computer programs to be configured to perform particular operations or actions means that the one or more programs include instructions that, when executed by data processing apparatus, cause the apparatus to perform the operations or actions. For example, a quantum computer may receive instructions from a digital computer that, when executed by the quantum computing apparatus, cause the apparatus to perform the operations or actions. Computers suitable for the execution of a computer program can be based on general or special purpose processors, or any other kind of central processing unit. Generally, a central processing unit will receive instructions and data from a read-only memory, a random access memory, or quantum systems suitable for transmitting quantum data, e.g. photons, or combinations thereof . The elements of a computer include a central processing unit for performing or executing instructions and one or more memory devices for storing instructions and digital, analog, and / or quantum data. The central processing unit and the memory can be supplemented by, or incorporated in, special purpose logic circuitry or quantum simulators. Generally, a computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information. However, a computer need not have such devices. Quantum circuit elements (also referred to as quantum computing circuit elements) include circuit elements for performing quantum processing operations. That is, the quantum circuit elements are configured to make use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data in a non-deterministic manner. Certain quantum circuit elements, such as qubits, can be configured to represent and operate on information in more than one state simultaneously. Examples of superconducting quantum circuit elements include circuit elements such as quantum LC oscillators, qubits (e.g., flux qubits, phase qubits, or charge qubits), and superconducting quantum interference devices (SQUIDs) (e.g., RF-SQUID or DC-SQUID), among others. In contrast, classical circuit elements generally process data in a deterministic manner. Classical circuit elements can be configured to collectively carry out instructions of a computer program by performing basic arithmetical, logical, and / or input / output operations on data, in which the data is represented in analog or digital form. In some implementations, classical circuit elements can be used to transmit data to and / or receive data from the quantum circuit elements through electrical or electromagnetic connections. Examples of classical circuit elements include circuit elements based on CMOS circuitry, rapid single flux quantum (RSFQ) devices, reciprocal quantum logic (RQL) devices and ERSFQ devices, which are an energy-efficient version of RSFQ that does not use bias resistors. In certain cases, some or all of the quantum and / or classical circuit elements may be implemented using, e.g., superconducting quantum and / or classical circuit elements. Fabrication of the superconducting circuit elements can entail the deposition of one or more materials, such as superconductors, dielectrics and / or metals. Depending on the selected material, these materials can be deposited using deposition processes such as chemical vapor deposition, physical vapor deposition (e.g., evaporation or sputtering), or epitaxial techniques, among other deposition processes. Processes for fabricating circuit elements described herein can entail the removal of one or more materials from a device during fabrication. Depending on the material to be removed, the removal process can include, e.g., wet etching techniques, dry etching techniques, or lift-off processes. The materials forming the circuit elements described herein can be patterned using known lithographic techniques (e.g., photolithography or e-beam lithography). During operation of a quantum computational system that uses superconducting quantum circuit elements and / or superconducting classical circuit elements, such as the circuit elements described herein, the superconducting circuit elements are cooled down within a cryostat to temperatures that allow a superconductor material to exhibit superconducting properties. A superconductor (alternatively superconducting) material can be understood as material that exhibits superconducting properties at or below a superconducting critical temperature. Examples of superconducting material include aluminum (superconductive critical temperature of 1.2 kelvin) and niobium (superconducting critical temperature of 9.3 kelvin). Accordingly, superconducting structures, such as superconducting traces and superconducting ground planes, are formed from material that exhibits superconducting properties at or below a superconducting critical temperature. In certain implementations, control signals for the quantum circuit elements (e.g., qubits and qubit couplers) may be provided using classical circuit elements that are electrically and / or electromagnetically coupled to the quantum circuit elements. The control signals may be provided in digital and / or analog form. Computer-readable media suitable for storing computer program instructions and data include all forms of non-volatile digital and / or quantum memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto- optical disks; CD-ROM and DVD-ROM disks; and quantum systems, e.g., trapped atoms or electrons. It is understood that quantum memories are devices that can store quantum data for a long time with high fidelity and efficiency, e.g., light-matter interfaces where light is used for transmission and matter for storing and preserving the quantum features of quantum data such as superposition or quantum coherence. Control of the various systems described in this specification, or portions of them, can be implemented in a computer program product that includes instructions that are stored on one or more non-transitory machine-readable storage media, and that are executable on one or more processing devices. The systems described in this specification, or portions of them, can each be implemented as an apparatus, method, or system that may include one or more processing devices and memory to store executable instructions to perform the operations described in this specification. While this specification contains many specific implementation details, these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. Certain features that are described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable sub-combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination. Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system modules and components in the implementations described above should not be understood as requiring such separation in all implementations, and 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. Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous. What is claimed is:

Claims

CLAIMS 1. A method performed by a quantum computer, the method comprising: determining a quantum circuit that performs unitary time evolution for a first Hamiltonian, comprising: mapping the first Hamiltonian to a corresponding gap-amplifiable Hamiltonian with an energy subspace specified by a parameter Δ, the gap-amplifiable Hamiltonian comprising one or more matrices, wherein each matrix comprises a respective block encoding that is dependent on a parameter ^^; determining an intermediate Hamiltonian, wherein eigenvalues of the intermediate Hamiltonian are the same as eigenvalues of a square root of the gap-amplifiable Hamiltonian; computing a block encoding of the intermediate Hamiltonian divided by the square root of ^^; and applying a filter to the block encoding of the intermediate Hamiltonian divided by the square root of λ to obtain a polynomial approximation of the unitary time evolution operator for the first Hamiltonian; and applying the quantum circuit to an initial state that is supported in the energy subspace specified by Δ of the gap-amplifiable Hamiltonian.

2. The method of claim 1, wherein the gap-amplifiable Hamiltonian is a Hamiltonian ^^with ground state energy ^^^ and low energy subspace ^^^^,^^^ ^ Δ^ such that ^^ െ ^^ ൌ∑^ ^ୀ^ ^^ற^^^^ , where ^^ ^ ^^^, ^^^ െ ^^ ൌ ^^^Δ^, and ^^^^^^^ୀ^represents the one or more matrices.

3. The method of claim 1 or claim 2, further comprising: obtaining the block encodings of the one or more matrices; or constructing the block encodings of the one or more matrices using an oracle model.

4. The method of claim 3, wherein the oracle model comprises a linear combination of unitaries model or a sparse model.

5. The method of any one of the preceding claims, wherein the quantum circuitcomprises ^^ ^ ^ఒ^queries to the block encoding of the intermediateHamiltonian divided by the square root of ^^, wherein each query is included in the quantum circuit and comprises one or more quantum logic gates.

6. The method of claim 5, wherein each query comprises a query to a controlled block-encoding unitary ^^, wherein the block-encoding unitary satisfies Π^^Π ൌ ୌି^^ି^ where Π represents a projector operator, ^^ represents the gap-amplifiable Hamiltonian ^^, and ^^ represents an energy that is less than or equal to the ground stateof the gap- amplifiable Hamiltonian.

7. The method of any one of the preceding claims , wherein the polynomial approximation is obtained over an interval ^െ^^,^^^, w^ఒ ఒ herein t represents time and ఒ^ 1.

8. The method of any one of the wherein computing the block encoding of the intermediate Hamiltonian divided by the square root of λ comprises performing spectral gap amplification.

9. The method of any one of the preceding claims, wherein applying the filter to the block encoding of the intermediate Hamiltonian divided by the square root of ^^ to obtain the polynomial approximation of the unitary time evolution operator for the Hamiltonian comprises performing a quantum singular value transform.

10. The method of any one of the preceding claims, wherein applying the quantum circuit to the initial state comprises: combining sine and cosine components of the polynomial approximation; applying a linear combination of unitaries transformation to the combined sine and cosine components to obtain a block encoding of the unitary time evolution operator for the first Hamiltonian; and compiling the quantum circuit according to the block encoding of the unitary time evolution operator for the first Hamiltonian.

11. The method of claim 10, wherein compiling the quantum circuit according to the block encoding of the unitary time evolution operator for the first Hamiltonian comprisesdecomposing unitary operators in the linear combination of unitaries transformation of the combined sine and cosine components to native quantum logic gates for the quantum computing device.

12. The method of any one of the preceding claims, wherein applying the quantum circuit to the initial state generates an evolved state, and wherein the method further comprises measuring the evolved state to obtain a measured result of the unitary time evolution.

13. An apparatus comprising:one or more classical processors; and one or more quantum computing devices in data communication with the one or more classical processors; wherein the apparatus is configured to perform the method of any one of claims 1 to 10.

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