Quantum rejection sampling for state preparation and matrix block encoding

Abstract

Quantum computing devices, systems and methods for using quantum rejection sampling to prepare a quantum register in a target quantum state or to block encode a target matrix to operate on a quantum register. A preparation operator is applied to a first quantum register and a second quantum register to prepare the first quantum register in a reference state and to prepare the second quantum register in a uniform register state. The reference state approximates the target quantum state and is a summation over a set of computational basis states weighted by a reference function. Rejection sampling is performed on the first quantum register to prepare the first quantum register in the target quantum state.

Description

7246-02201PsiQ-563W01QUANTUM REJECTION SAMPLING FOR STATE PREPARATION AND MATRIX BLOCK ENCODINGTechnical Field|001] Embodiments herein relate generally to quantum computational methods, systems and devices for performing quantum state preparation.Background

[0002] Quantum computing can be distinguished from “classical'’ computing by its reliance on structures referred to as “qubits.” At the most general level, a qubit is a quantum system that may exist in one of two orthogonal states (denoted as 10) and 1) in the conventional bra / ket notation) or in a superposition of the two states (e.g.. -)=(|0) + 11». BY operating on a system (or ensemble) of qubits, a quantum computer may quickly perform certain categories of computations that would require impractical amounts of time in a classical computer.

[0003] One application of quantum computing is the simulation of physical quantum systems. The quantum system may include a plurality of particles and / or fields of different types, with differing properties and interactions. Quantum simulation involves preparing qubits into an initial quantum state, which may be a complex and computationally intensive procedure, particularly tor more complex systems. Accordingly, improvements in the field of quantum state preparation are desired to increase the efficiency and reduce the complexity of quantum simulation and other quantum computational methods.Summary

[0004] Embodiments described herein include quantum computing devices, systems, quantum circuits, and methods for performing quantum state preparation to prepare a quantum register in a target quantum state or to block encode a target matrix to operate on a quantum register.

[0005] In some embodiments, a preparation operator is applied to a first quantum register and a second quantum register to prepare the first quantum register in a reference state and to prepare the second quantum register in a uniform register state. In some embodiments, the reference state is entangled with the uniform register state, lire reference state is a summation7246-02201PsiQ-563W01over a set of computational basis states weighted by a reference function, and the target quantum state is a summation over the computational basis states weighted by an amplitude function. In some embodiments, the reference function is an approximation of the amplitude function is greater than or equal to the amplitude function in magnitude for corresponding computational basis states.[0061 In some embodiments, rejection sampling is performed on the first quantum register to prepare the first quantum register in the target quantum state. Performing rejection sampling involves rejecting samples from the first quantum register with a magnitude of the reference function that exceeds a magnitude of the amplitude function.1007i The techniques described herein may be implemented in and / or used with a number of different types of devices, including but not limited to photonic quantum computing devices and / or systems, hybrid quantum / classical computing systems, and any of various other quantum computing systems.[0081 This Summary is intended to provide a brief overview of some of the subject matter described in this document. Accordingly, it will be appreciated that the above-described features are merely examples and should not be construed to narrow the scope or spirit of the subject matter described herein in any way. Other features, aspects, and advantages of the subject matter described herein will become apparent from the following Detailed Description, Figures, and Claims.Brief Description of the Drawings10091 For a better understanding of the various described embodiments, reference should be made to the Detailed Description below', in conjunction with the following drawings in w'hich like reference numerals refer to corresponding parts throughout the Figures.

[0010] Figure 1 A is a system diagram illustrating a quantum computing system, according to some embodiments;

[0011] Figures 1B-G illustrate the utilization of surface codes to constructed an error-corrected fault-tolerant logical qubit, according to some embodiments;

[0012] Figure 2 is a flowchart diagram illustrating a method for performing quantum state preparation, according to some embodiments;

[0013] Figure 3 is a flowchart diagram illustrating a method for performing matrix block encoding, according to some embodiments;7246-02201PsiQ-563W01

[0014] Figure 4 is a quantum circuit diagram configured to perform quantum state preparation, according to some embodiments;

[0015] Figure 5 is a quantum circuit diagram illustrating a subroutine for encoding the application of a complex phase, according to some embodiments;

[0016] Figure 6 is a quantum circuit diagram illustrating a subroutine for performing a PREPg operation, according to some embodiments;

[0017] Figure 7 is a quantum circuit diagram illustrating a subroutine tor preparing a piecewise -constant quantum state, according to some embodiments;

[0018] Figure 8 is a quantum circuit diagram illustrating a method for performing matrix block encoding, according to some embodiments;

[0019] Figure 9 is a quantum circuit diagram illustrating a matrix for block encoding a matrix using submatrix partitioning, according to some embodiments;

[0020] Figure 10 illustrates a matrix ziggurat with four regions, according to some embodiments;

[0021] Figure 11 is a quantum circuit diagram illustrating a matrix for block encoding a matrix using row-column encoding, according to some embodiments;

[0022] Figure 12 is a quantum circuit diagram illustrating a matrix for block encoding a matrix using column encoding, according to some embodiments;

[0023] Figure 13 is a quantum circuit diagram configured to perform state preparation with modified functions f and g, according to some embodiments;

[0024] Figure 14 is a quantum circuit diagram configured to perfonn state preparation where the comparator has been replaced with a coherent computation of angles, according to some embodiments;

[0025] Figure 15 is a quantum circuit diagram configured to perform matrix block encoding where the comparator has been replaced with a coherent computation of angles, according to some embodiments;

[0026] Figure 16 is a quantum circuit diagram illustrating a decomposition of a time evolution operator into a plurality of qubitization operators, according to some embodiments; and

[0027] Figure 17 is a quantum circuit diagram illustrating the component subroutines of a qubitization operator, according to some embodiments.

[0028] While the features described herein may be susceptible to various modifications and alternative forms, specific embodiments thereof are shown by way of example in the drawings7246-02201PsiQ-563W01and are herein described in detail. It should be understood, however, that the drawings and detailed description thereto are not intended to be limiting to the particular form disclosed, but on the contrary, the intention is to cover all modifications, equivalents and alternatives falling wi thin the spirit and scope of the subject matter as defined by the appended claims.DETAILED DESCRIPTION

[0029] Disclosed herein are examples (also referred to as “embodiments”) of systems and methods for simulating a physical quantum system using various quantum computing systems, including photonic systems.

[0030] Although embodiments are described with specific detail to facilitate understanding, those skilled in the art with access to this disclosure will appreciate that the claimed invention may be practiced without these details. Reference will now be made in detail to embodiments, examples of which are illustrated in the accompanying drawings. In other instances, well-known methods, procedures, components, circuits, and networks have not been described in detail so as not to unnecessarily obscure aspects of the embodiments.Overview of Quantum Computing

[0031] To facilitate understanding of the disclosure, an overview of relevant concepts and terminology is provided in the following paragraphs.

[0032] Quantum computing relies on the dynamics of quantum objects, e.g., photons, electrons, atoms, ions, molecules, nanostructures, and the like, which follow the rules of quantum theory'. In quantum theory, the quantum state of a quantum object is described by a set of physical properties, the complete set of which is referred to as a mode. In some embodiments, a mode is defined by specifying the value (or distribution of values) of one or more properties of the quantum object. For example, in the case where the quantum object is a photon, modes may be defined by the frequency of the photon, the position in space of the photon (e.g., which waveguide or superposition of waveguides the photon is propagating within), the associated direction of propagation (e.g., the ^-vector for a photon in free space), the polarization state of the photon (e.g., the direction (horizontal or vertical) of the photon’s electric and / or magnetic fields), a time window in which the photon is propagating, the orbital angular momentum state of the photon, and the like.

[0033] Persons of ordinary skill in the art will be able to implement examples using any of a variety of types of quantum systems, including but not limited to photonic systems, solid state7246-02201PsiQ-563W01system, topological quantum computing systems, hybrid quantum computing systems, and superconducting systems, among other possibilities.

[0034] As used herein, a “qubit” (or quantum bit) is a quantum system with an associated quantum state that may be used to encode information. A quantum state may be used to encode one bit of information if the quantum state space can be modeled as a (complex) two- dimensional vector space, with one dimension in the vector space being mapped to logical value 0 and the other to logical value 1. In contrast to classical bits, a qubit may have a state that is a superposition of logical values 0 and 1. More generally, a “qudit” describes any quantum system having a quantum state space that may be modeled as a (complex) n-dimensional vector space (for any integer i?), which may be used to encode log2(n) bits of information. For the sake of clarity of description, the term “qubit” is used herein, although in some embodiments the system may also employ quantum information carriers that encode information in a manner that is not necessarily associated with a binary bit, such as a qudit or a plurality of qubits encoded to form an error-corrected logical qubit. For exampl e, embodiments herein for quantum computational methods and circuits that utilize fault- tolerant quantum computing schemes use the tenn “qubit” to refer to an error-corrected logical qubit that contains a plurality of physical qubits entangled together in an error¬ correcting code.

[0035] Embodiments herein use the term “quantum register” to refer to a set of one or more logical qubits used in a quantum computational method. Typically, distinct quantum registers are separated at the logical layer, i,e., different quantum registers serve distinct logical purposes in a quantum computing method or circuit.036 [ Qubits (or qudits) may be implemented in a variety of quantum systems. Examples of qubits include: polarization states of photons; presence of photons in waveguides; or energy states of molecules, atoms, ions, nuclei, or photons. Other examples include other engineered quantum systems such as flux qubits, phase qubits, or charge qubits (e.g., formed from a superconducting Josephson junction); topological qubits (e.g., Majorana fermions); or spin qubits formed from vacancy centers (e.g,, nitrogen vacancies in diamond).Figure 1A - Quantum Computing System[037 j Figure 1A is a system diagram of a quantum computing system 101 that may be utilized to implement method steps of embodiments described herein. As illustrated, the system includes a classical computing system 103 coupled to a quantum processing unit7246-02201PsiQ-563W01(QPU) 105 over a classical channel 112. The classical channel may relay classical information between the classical computing sy stem and the QPU.

[0038] In some embodiments, the classical computing system 103 includes one or more non- transitory computer-readable memory media 104, one or more central processing units (CPUs) or processor(s) 102, a power supply, an input / output (I / O) subsystem, and a communication bus interconnecting these components. The processor(s) 102 may execute modules, programs, and / or instructions stored in memory' 104 and thereby perform processing operations. The processor(s) may additionally or alternatively perform operations based on information and / or instructions received from the QPU 105 over the channel 112. The processor may comprise a dedicated processor, or it may be a field programmable gate arrays (FPGA), an application specific integrated circuit (ASIC), or a “system on a chip” that includes classical processors and memory, among other possibilities. In some embodiments, memory 104 stores one or more programs (e.g., sets of instructions) and / or data structures and is coupled to the processor(s).

[0039] The classical computing system may be classical in the sense that it operates computer code represented as a plurality of classical bits that may take a value of 1 or 0. Programs may be written m the form of ordered lists of instructions and stored within the classical (e.g., digital) memory 104 and executed by the classical (e.g., digital) processor 102 of the classical computer. The memory 104 is classical in the sense that it stores data and / or program instructions in a non-transitory storage medium in the form of bits, which have a single definite binary state at any point in time. The processor may read instructions from the computer program in the memory' 104 and / or write data into memory, and may optionally receive input data from a source external to the computer 103, such as from a user input device such as a mouse, keyboard, or any other input device. The processor 102 may execute program instructions that have been read from the memory 104 to perform computations on data read from the memory 104 and / or input from the QPU, and generate output from those instructions. The processor 102 may store that output back into the memory 104 and / or provide the output to the QPU over the channel 112.

[0040] The QPU 105 may include a plurality of qubits 110 and a controller 106 configured to interface with the plurality of qubits. In some embodiments, the qubits are divided into one or more independent qubit modules, where each qubit module includes a self-contained plurality of fault-tolerant qubits, and different qubit modules may be interchangeably used for various steps within a quantum computation. The controller 106 may include physical hardware to7246-02201PsiQ-563W01interact with and / or perform operations on the qubits, e.g., to apply quantum gates or perform other operations. In some embodiments, the controller further includes a classical processor, potentially coupled to its own dedicated non-transitory (classical) memory', that is configured to direct the physical hardware to interact with the qubits and communicate with the processor of the classical computing system 103 over the channel 112. Alternatively, the classical processor of the classical computing system 103 may directly communicate with the hardware of the controller to provide instructions for interacting with and manipulating the qubits. The qubits may be configured to evolve in time under the directed influence of the controller, and a measurement system 108 may at times perform quantum measurements on all or a subset of the qubits to obtain quantum measurement results in the form of classical data bits (e.g., ones and zeros). Tire classical data from the measurement results may be intermediate results that inform behavior of the classical computing system and / or the quantum controller 106 during a quantum computation, and they may additionally include classical results of the quantum computation. In some embodiments, the QPU further includes one or more decoders configured to receive and decode the classical measurement results, and the decoded measurement results may be provided to the classical computing system for processing. The measurement results may be communicated to the classical computing system and / or the controller 106, and further the classical computing system may provide directions and / or instructions to the controller 106 and the measurement system 108 to guide the behavior of the QPU while performing a quantum computation.Figures 1B-G - Surface Codes and Physical implementations[041 J Qubits (and operations on qubits) may be implemented using a variety of physical systems. In some embodiments, qubits are provided m an integrated photonic system employing waveguides, beam splitters, photonic switches, and single photon detectors, and the modes that may be occupied by photons are spatiotemporal modes that correspond to presence of a photon in a waveguide. Modes may be coupled using mode couplers, e.g., optical beam splitters, to implement transformation operations, and measurement operations may be implemented by coupling single-photon detectors to specific waveguides. One of ordinary skill in the art with access to this disclosure will appreciate that modes defined by any appropriate set of degrees of freedom, e.g., polarization modes, temporal modes, and the like, may be used without departing from the scope of the present disclosure. For instance, for modes that only differ in polarization (e.g., horizontal (H) and vertical (V)), a mode7246-02201PsiQ-563W01coupler may be any optical element that coherently rotates polarization, e.g., a birefringent material such as a waveplate. For other systems such as ion trap systems or neutral atom systems, a mode coupler may be any physical mechanism that couples two modes, e.g,, a pulsed electromagnetic field that is tuned to couple two internal states of the atom / ion.10421 In some embodiments of a photonic quantum computing system rising dual-rail encoding, a qubit may be implemented using a pair of waveguides. In some embodiments, a photon in a first waveguide of the pair and no photon in a second waveguide of the pair (also referred to as a vacuum mode) may correspond to the i 0) state of a photonic qubit.Alternatively, a state with a photon in the second waveguide and no photon in the first waveguide may correspond to the 1 ) state of the photonic qubit. To prepare a photonic qubit in a known logical state, a photon source may be coupled to one end of one of the waveguides. The photon source may be operated to emit a single photon into the waveguide to which it is coupled, thereby preparing a photonic qubit in a known state. Photons travel through the waveguides, and by periodically operating the photon source, a quantum system having qubits whose logical states map to different temporal modes of the photonic system may be created in the same pair of waveguides. In addition, by providing multiple pairs of waveguides, a quantum system having qubits whose logical states correspond to different spatiotemporal modes may be created. It should be understood that the waveguides in such a system need not have any particular spatial relationship to each other. For instance, they may be but need not be arranged in parallel,[0431 Some embodiments described below relate to physical implementations of unitary operations that couple modes of a quantum system, which may be understood as transforming the quantum state of the system. For instance, if the initial state of the quantum system (prior to mode coupling) is one in which one mode is occupied with probability 1 and another mode is unoccupied with probability 1 (e.g., a state 10} in Fock notation), mode coupling may result in a sta te in which both modes have a nonzero probability of being occupied, e.g., a state a1|10⟩ + a2|01⟩, where |a1|2+ |a2|2= 1. In some embodiments, operations of this kind may be implemented by using beam splitters to couple modes together and variable phase shifters to apply phase shifts to one or more modes. The amplitudes ai and az depend on the reflectivity (or transmissivity) of the beam splitters and on any phase shifts that are introduced.[0441 A single physical qubit (e.g., such as the 2-level physical qubit illustrated in Figure IB with a quantum state~ a 0) + a2|1⟩) may be used for quantum computation in7246-02201PsiQ-563W01principle. However, individual physical qubits are generally highly susceptible to noise and decoherence. Fault-tolerant quantum computing utilizes a plurality of entangled physical qubits to encode a single logical qubit to mitigate the frailty and / or short coherence times of individual physical qubits. In fault-tolerant quantum computing schemes, a plurality of physical qubits is entangled together according to a specific error-correcting code (e.g., using fusion measurements on resource states) to produce a single logical qubit that is less susceptible to noise and decoherence, such as is shown in Figure 1C.10451 Figure 1C illustrates one example for constructing a fault-tolerant logical qubit using a circuit-based approach. In the illustrated example, the light shaded circles are data qubits (e.g., qubits 125-131) that encode quantum information. The data qubits are entangled with adjacent measure qubits, illustrated as dark shaded circles (such as measure qubit 123). The measure qubits may be measured to determine aspects of the quantum information encoded in the data qubits. The example illustrated in Figure 1C has a code length of d ~ 12. Fusion¬ based approaches to encoding fault-tolerant logical qubits may also be used for embodiments described herein. Encoding qubits in this manner causes the resultant logical qubit to be less sensitive to error and noise, and resultant errors may be fixed via quantum error correction. Encoding a logical qubit may itself be vulnerable to errors, which may likewise be corrected and / or tolerated.

[0046] In some quantum computing methodologies, such as fusion-based quantum computing, a logical qubit is encoded from a plurality of physical qubits using a sequence of specific measurements (e.g., stabilizer measurements). The measurement sequence may be constructed where a subset of the physical qubits is measured (e.g., collapsing the quantum state and producing classical information, i.e., the measurement result) in such a way that the remaining unmeasured / un-collapsed degrees of freedom (e.g., a 2-dimensional subspace which has support over all the phy sical qubits) form tire desired encoded logical qubit.Accordingly, the processes of performing stabilizer measurements and / or encoding a fault- tolerant logical qubit may receive a plurality of physical qubits as input and as output may produce both the encoded logical qubit and classical information (e.g., syndrome graph data) resulting from the measurement sequence.

[0047] In some embodiments, a logical qubit may be a component of a quantum error¬ correcting code where an operation (for example, a quantum gate acting on the logical qubit) may be performed on encoded logical information. For example, a logical qubit may include multiple resource states that are entangled with one another in a specific way. Resource states7246-02201PsiQ-563W01are defined as a plurality of physical qubits prepared in a specific entangled manner. In some embodiments, 6-qubit resource states may be used, or other types of resource states may be used.1048 i If the above-described surface code measurement schedule is applied for numerous time steps, the system effectively acts as a fault-tolerant quantum memory for the logical qubit encoded by the underlying surface code or, viewed another way, as a fault-tolerant logical identity gate on the logical qubit that is encoded by the underlying surface code. Viewed yet another way, this process operates as a fault-tolerant logical channel.049 [ Figure ID illustrates a 3-dimensional graphical depiction of such a fault-tolerant logical identity gate. The surface labeled 114 is the input to the gate and includes an arbitrary logical state encoded in a surface code, represented as the input checkerboard surface.Likewise, the surface labeled 118 identifies the output qubits after the identity gate / has been applied to it. As one example, in a circuit-based implementation the fault-tolerant logical qubit shown in Figure 1C may be utilized as the input surface 114, which may be operated on within the illustrated volume and output as the surface 118. The input and output surfaces, which may be associated with either the physical or relational arrangement of qubits, are connected to each other via an intervening volume that represents the unique set of measurements to be applied over time. Accordingly, in Figure ID, time flow's from left to right and the lighter shaded (front and back) and darker shaded (top and bottom) sides of the boundaries of the volume depict whether the primal or dual plaquettes are disposed on that boundary. Figure IE represents the same concept but written in a more familiar quantum circuit notation illustrating the analogy between the more familiar quantum circuit. While Figure ID shows the logical identity gate, any gate can be depicted in this manner and such a depiction is one example of a logical block that specifies a set of instructions to be performed on the underlying surface code qubits to perform a logical operation (the identity gate in this example) on the logical qubit that is encoded by surface code. Other examples of such gates are the S gate, the Hadamard gate, and the CX gate, among other possibilities. This combination of gates may be used, for example, to implement the quantum circuits described in various embodiments.

[0050] The protocol for preparing an encoded logical state may contain two parameters, L and La. Here L is referred to as the “distance” of the scheme, which corresponds to the length and width of the cross section shown in Figure ID - it determines the code distance of the surface code state being prepared. In some embodiments, L may be separated into two7246-02201PsiQ-563W01parameters, Lxand Ly, i.e., the code distance may be different in the two spatial directions. This may be desirable, for example, when there is an asymmetry in the noise model or logical error rates in the X and Z directions, and the code distance may be separately tuned in the two spatial directions. Ldis referred to as the “depth” of the scheme - it can be thought of as simulated time, i.e. the number of rounds of stabilizer measurements in CBQC, or the number of layers of resource states in FBQC.Amay de termine the number of stabilizer checks in the protocol from which information may be gathered for post-selection. A minimal depth of LA= 2 may be chosen, however, longer depths may also be used (using more overhead) to allow for more information to be collected in order to better predict logical errors on the output state.

[0051] Figures IF and 1G illustrate a specific example in fusion-based quantum computing (FBQC) of an arrangement of physical qubits that may be used to perform a (Z2, ZJ) measurement on four logical qubits qi-q4. The individual circles shown in the rectangular sheet 120 in the top half of Figure 1G represent individual physical qubits, and the lines connecting adjacent qubits indicate entanglement (e.g., via fusion measurements). In the stack oft / = 9 layers shown at 122 of Figure 1G, the vertical direction represents the depth of the logical qubit (i.e., time), which is a sequence of nine entangling measurements performed on the 9x9 grid of physical qubits representing each of the qubits qi-q4 as well as a portion of the auxiliary qubits 121. While Figures IF and 1G illustrate an arrangement of physical qubits that may be used to perform a simple dual-qubit measurement, physical qubits may be configured as quantum registers in more complex arrangements to perform the operations shown in the quantum circuit diagrams shown in Figures 4-9 and 11-15.Quantum Rejection Sampling for Quantum State Preparation and Matrix Block Encoding

[0052] Embodiments herein describe quantum computing systems, quantum circuits, and methods for using quantum rejection sampling to prepare a quantum register into a particular quantum state or for block encoding a matrix to operate on a quantum register. The described methods may be used in various contexts, for example, for initial state preparation or data- loading as a subroutine for general quantum computation,

[0053] The quantum state may describe the state of a plurality of particles of a physical system, which may include one or more molecules, atoms, substances, etc. In some embodiments, the particles include reactants and / or catalysts of a chemical reaction, and the state describes an initial state for the chemical reaction. In some embodiments, the quantum state is a thermal7246-02201PsiQ-563W01quantum state that describes electronic and / or thermal (i.e., phononic or vibratory) degrees of freedom of the particles of the physical system.

[0054] Expressed in explicit mathematical form, some embodiments herein describe methods and computing systems that, given a family of sets of complex numbers { / (x) 6 with increasing N ∈ N+, prepare a quantum register in the quantum state:

[0055] (1) J

[056] whereanormalization factor.

[0057] To prepare the quantum register in tire target state, embodiments herein design the initial reference state according to the target state. Namely, an initial reference state may be selected of the following form, which may be efficiently generated by a unitary operation:

[0058] ;M-HO. (2)

[0059] Unitary operators Uy. lx) ® |0) i-» x) ® / f(x) (and similarly f may be used that are computationally efficient to implement, f and g are functions that are related to the original functions f and g, respectively. They can be chosen in various ways and may be designed such that state preparation may be performed with only 0(1) rounds of amplitude amplification. Some embodiments use an ancilla register of dimension M that depends on tire choice of / ', g and the desired precision of the target state.

[0060] More formally, quantum state preparation may be defined as follows.

[0061] Problem 1.1 (Quantum state preparation) Let {f'(x)E be a family of sets of complex numbers with increasing N ∈ N+. The quantum state preparation problem is the problem, of producing the quantum state!qp) that is an e -approximation to the target state i0r):” 77-Sr-i be.,1Jv f

[0062] || | / >) - i / y> ||< e (3)

[0063] where ||- 1| is the Euclidean norm, and 0 < E < 1.

[0064] Some embodiments address the related problem of block-encoding a matrix in a quantum register, i.e., encoding a given matrix into a submatrix of a unitary operator. The problem of matrix block-encoding may be defined as follows:7246-02201PsiQ-563W01

[0065] Problem 1.2 (Matrix block-encoding) Let {4j7- ebe a family of sets of complex numbers with increasing M, N ∈ ℕ+. The matrix block-encoding problem is the problem of producing a unitary U that embeds an e-approximation to the a-rescaled versionof the matrix A’. = Ay I0GI- be.,

[0066] H (x | U 0)t-.4 / a ii < c (4)

[0067] where EE-II is the operator norm, 0 < e: < 1, |0)aidentifies the subspace of the ancilla system where the approximation of 4 is embedded, and a ∈ ℝ+is a rescaling factor.Figure 2 -- Flowchart for Quantum State Preparation

[0068] Figure 2 is a flowchart diagram illustrating systems, quantum computing devices, and methods for utilizing quantum rejection sampling to prepare a quantum register in a target state, according to some embodiments. Quantum rejection sampling prepares a quantum register in a target state by first preparing the quantum register in a simpler reference state (i.e., computationally simpler to prepare), and coherently rejecting samples from this state that differ from the target state until the quantum register approaches the target state. The method shown in Figure 2 may be used in conjunction with any of the computer systems or devices shown in the above Figures, among other devices. For example, the method shown in Figure 2 may be performed by a quantum computing device or system 101 as illustrated in Figure 1A. The quantum computing system may include a quantum processing unit 105 configured to direct the described method steps that is coupled to a classical computing system 103 for processing classic information and directing operations of the quantum processing unit. In some embodiments, a quantum controller 106 may apply a sequence of gates to the qubits 110 under the direction of the classical computing system 103 to prepare the qubits into the target quantum state.

[0069] In some embodiments, the methods described in Figure 2 may be implemented using the quantum circuit illustrated in Figure 4. In the description of Figure 2, reference numerals in parentheses are used to refer to corresponding elements of the quantum circuit diagram shown in Figure 4.

[0070] In some embodiments, the described quantum circuits may be implemented in any of a variety of types of quantum computing systems, including but not limited to photonic, semiconductor, superconducting and / or topological quantum computing systems. It is to be7246-02201PsiQ-563W01understood this method may be used by any of a variety of types of quantum computing architectures, and these other types of systems should be considered within the scope of the embodiments described herein. As illustrated, the method shown in Figure 2 may proceed as follows.

[0071] At 202, a preparation operator (420) is applied to a first quantum register and a second quantum register to prepare the first quantum register in a reference state (412) and to prepare the second quantum register in a uniform register state (414). The preparation operator may consist of two subroutines, PREPq(as defined in Definition 2.1, below) to prepare the first quantum register in the reference state \ibc,) ~ ( ) X), and USPM(as defined in' figDefinition 2.4, below) to prepare the second quantum register in the uniform superposition register state \USPM') — The concatenated operation PREP„ ® USPMmay beapplied to the first and second quantum registers so that the collective state of the first and second quantum registers is the product state | [ ) = (gl / SPMafter application of the preparation operator. Tire reference state is a summation over a set of computational basis states |x) weighted by a reference function g(x'). The reference function g(x) is a bounding function that resembles the absolute value (or magnitude) of the amplitude function if( ) of a target quantum state. Note that g(x) may be selected to have positive and real values for each value of x, whereas f (x) may have negative and / or complex values, in some embodiments.

[0072] Note that “x”, as an argument of the function f (x) or g x), refers to an integer value in the range of (1, 2,..., N}. This is distinct from |x), which refers to a quantum binary' encoding of the integer value x in a quantum register as a quantum state, ix) may also be referred to as the computational basis representation of the integer x. One possible encoding is, when N = 2n, |x) may be encoded with n qubits, where |x = 0) is encoded as0) 0)... |0), |x = 1} is encoded as |0) 10)... 11), x = 2) is encoded as 0) 10)... |l)|0), etc., as one non-limiting example.

[0073] In some embodiments, the reference function is a piecewise constant function, a piecewise linear function, an exponential function, a piecewise exponential function, or a combination thereof. For example, in some embodiments, the reference function may be a piecewise function that is a combination of linear and exponential domains. In general, the reference function may be determined based on the amplitude function of the target quantum state to approximate the amplitude function as closely as possible (being greater than or equal7246-02201PsiQ-563W01to the amplitude function for all values of x) while also being computationally simple to prepare in a quantum register. When the reference function is a piecewise linear function, the piecewise linear function may share values with one or more maxima of the amplitude function. When the reference function is an exponential function, the exponential function may track a Gaussian profile or another profile of the amplitude function. Several specific examples of the reference function are described in Examples 1-3, below, according to at least some embodiments. In addition, specific examples for constructing the reference function when the target function includes either a power law dependency or a Gaussian profile are described in greater detail below.

[0074] In some embodiments, the piecewise linear function is determined by dividing a domain of the amplitude function into plurality of subdomains, and, within each subdomain, identifying a respective magnitude of the piecewise linear function as the maximum magnitude of the amplitude function within the respective subdomain. As one example, the amplitude function may be divided into a separate subdomain for regions surrounding each peak of the amplitude function, and the reference function may be determined to have a magnitude equal to the peaks in each respective subdomain.

[0075] The target quantum state (416) includes a summation over the computational basis states x) weighted by the amplitude function f (x). The magnitude of the reference function is greater than or equal to the magnitude of the amplitude function / '(x) | for corresponding values of x. Said another way, for each value of x, the reference function is greater than or equal to the amplitude function in magnitude.

[0076] At 204, rejection sampling is performed on the first quantum register to prepare the first quantum register in the target quantum state. Performing rejection sampling involves rejecting samples from the first quantum register that have a magnitude of the reference function that exceeds a magnitude of the amplitude function. As used herein, “samples” refers to the separate terms in the summation over m in the product state&(x) lx) lmX Said another way, each sample is a termff(x)!x> “F= 2...., M}. As described in greater detail below, rejection. Kg ’ ’ VAfsampling involves rejecting samples corresponding to values of m for which the value of #(x)m is greater than \f(x)\M, which is checked for each value of m and for each value of x. Performing rejection sampling may include one or more of the subroutines (422), (424), (426), (428), (430) and (432) illustrated in Figure 4, in various embodiments.7246-02201PsiQ-563W01

[0077] In some embodiments, performing rejection sampling includes applying a first unitary operator Uf(422) and a second unitary operator Ug(424). The first unitary operator encodes the amplitude function of the target quantum state into a third quantum register (408). The second unitary operator (424) is applied to encode the reference function of the reference state into a fourth quantum register (410). These unitary operations are defined m Definition 2.2, below.

[0078] In some embodiments, a comparator operator Comp (426) may then be applied to entangle the first, third and fourth quantum registers with a flag register qubit (406). The comparator operator is defined in Definition 2.3, below. Applying the comparator operator to entangle tire second, third and fourth quantum registers coherently flags samples from the fourth quantum register based on a comparison of magnitudes of the reference function and magnitudes of the amplitude function in the third quantum register for corresponding values of x. The output state of the comparator operator is described explicitly in reference to Eq. (7), where for each value in the sample space of m = {1,the |0) state of the flag register qubit is entangled with samples of the fourth quantum register with |f(x) | M > g(x)m, whereas the 1) state of the flag register qubit is entangled with samples of the fourth quantum register with f(x)|M < g(x')m. The samples of the fourth quantum register that are entangled with the 0) state of the flag register qubit are in the desired subspace, and may be amplified in the subsequent step.

[0079] Finally, an amplitude amplification operation (428) may be applied to amplify the flagged samples from the fourth quantum register. The amplitude amplification operation projects the overall state |3) in Eq. (7) onto the |0) subspace of the flag register qubit. The result of the amplitude amplification operation is shown in Eq. (8), The amplitude amplification operation may include iteratively calling an amplification subroutine one or more times. The number of calls,,4, of the amplification subroutine may be determined based on aspects of the reference function and the amplitude function, as shown in Eq, 10. For example, increasing the number of rounds of amplitude amplification may increase the probability of successfully preparing the target quantum state at the cost of a higher computational load, and A may be selected to balance these two aspects.

[0080] In some embodiments, after applying the amplitude amplification operation, an inverse of the preparation operator, an inverse of the comparator operator, and an inverse of the first unitary operator are applied to the quantum registers (430), to return the second and7246-02201PsiQ-563W01third quantum registers to the null state and un-entangle them from the first quantum register. The first quantum register will then be in the desired target state ii / tf|), up to a phase factor.

[0081] In some embodiments, after applying the amplitude amplification operation, a respective phase for each eigenstate of the target quantum state is determined. In these embodiments, respective qubits of the first quantum register may be encoded with the respective determined phases. A quantum circuit illustrating an example method for determining the phase is shown in Figure 5.

[0082] In some embodiments, after preparing the first quantum register in the target quantum state, time evolution may be emulated for the target quantum state to emulate a self¬ thermalization process on the target quantum state.

[0083] The first quantum register may then be utilized while prepared in the target quantum state. For example, the first quantum register may be provided to a subsequent step in a quantum computation. As one example, in some embodiments the target quantum state may be used to construct a purified thermal state, and the purified thermal state may be used in a quantum simulation (e.g., of a chemical reaction involving the purified thermal state). In some embodiments, the first quantum register may be partially or entirely measured to obtain classical measurement results indicating the target quantum state.Figure 3 - Flowchart for Matrix Block Encoding

[0084] Figure 3 is a flowchart diagram illustrating systems, quantum computing devices, and methods for utilizing quantum rejection sampling to block encode a matrix to operate on a quantum register, according to some embodiments. Quantum rejection sampling block encodes a matrix to operate on a quantum register by block encoding a simpler reference matrix (i.e., computationally simpler to prepare), and coherently rejecting samples from this state that differ from the target state until the quantum approaches the target matrix. The method shown in Figure 3 may be used in conjunction with any of tire computer systems or devices shown in the above Figures, among other devices. For example, the method shown in Figure 3 may be performed by a quantum computing device or system 101 as illustrated in Figure 1A. The quantum computing system may include a quantum processing unit 105 configured to direct the described method steps that is coupled to a classical computing system 103 for processing classic information and directing operations of the quantum processing unit. In some embodiments, a quantum controller 106 may apply a sequence of7246-02201PsiQ-563W01gates to the qubits 110 under the direction of the classical computing system 103 to prepare the qubits into the target quantum state.

[0085] In some embodiments, the methods described in Figure 3 may be implemented using the quantum circuit illustrated in Figure 8. In the description of Figure 3, reference numerals in parentheses are used to refer to corresponding elements of the quantum circuit diagram shown in Figure 8.

[0086] In some embodiments, the described quantum circuits may be implemented in any of a variety of types of quantum computing systems, including but not limited to photonic, semiconductor, superconducting and / or topological quantum computing systems. It is to be understood this method may be used by any of a variety of types of quantum computing architectures, and these other types of systems should be considered within the scope of the embodiments described herein. As illustrated, the method shown in Figure 3 may proceed as follows.

[0087] At 302, a first quantum register is prepared in an initial state. The initial state may be null state, in some embodiments. Note that each of the three rails (816 a), (816 b), and (816 c) illustrated in Figure 8 are part of the first quantum register.

[0088] At 304, a preparation operator is applied to the first quantum register to block encode a reference matrix G to operate on the first quantum register. In some embodiments, the reference matrix is an approximation of a target matrix A. The reference matrix may be selected such that entries of the reference matrix have a magnitude that is greater than or equal to the magnitude of entries of the target matrix for corresponding indices.

[0089] In some embodiments, each entry of the reference matrix is a product of a first vector and a second vector. For example, the reference matrix may be a summation over a product %• -i • ■ X ' i i i or a first, tensor, yw / , and a second tensor, i.e.. Xtyi where t and / denotethe indices of the G. In these embodiments, applying the preparation operator may include a separate application of both a PREP operator and a PREP, operator, or a PREP^ operator and a PREP^ operator. As shown in Figure 8, in some embodiments the PREPXmay be applied prior to performing rejection sampling and PREP^ may be applied after, or vice versa.7246-02201PsiQ-563W01

[0090] At 306, rejection sampling is performed on the first quantum register to apply a block encoded approximation of the target matrix onto the first quantum register. Performing rejection sampling may include rejecting samples from the first quantum register with values tor entries of the reference matrix that exceed corresponding values of the target matrix. In some embodiments, performing rejection sampling includes one or more of the steps (804), (806), (808), (810) and (812) shown in Figure 8.

[0091] In some embodiments, performing rejection sampling includes applying a first unitary operator UG (804) to encode entries of the reference matrix into a second quantum register (818) and applying a second unitary operator (806) to encode entries of the target matrix into a third quantum register (820). A comparator operator (808) may then be applied to entangle the second and third quantum registers with a flag register qubit (822). Applying the comparator operator to entangle the second and third quantum registers flags samples from the third quantum register with magnitudes of the entries of the reference matrix that are less than or equal to a maximum magnitude of the entries of the target matrix in the second quantum register for corresponding indices.092] In some embodiments, after applying the comparator operator, an inverse of the first unitary operator (810) and an inverse of the second unitary operator (812) are applied to return the second and third quantum registers to a null state.

[0093] In some embodiments, the reference matrix is divided into a plurality of groups, wherein each group includes one or more cells with a respective constant value. These embodiments are described in greater detail in Figures 9-10 and the associated description related to matrix ziggurat submatrix partitioning.

[0094] In some embodiments, the preparation operator is a unitary operator that outputs a summation over square roots of entries of each row or column of the reference matrix. These embodiments are described in greater detail in Figure 11 and the associated description related to row-column block encoding.

[0095] In some embodiments, the preparation operator is a unitary operator that outputs a Frobenius norm of the reference matrix for each row or column of the reference matrix. These embodiments are described in greater detail in Figure 12 and the associated description related to column block-encoding.7246-02201PsiQ-563W01

[0096] In some embodiments, a respective phase for each entry of the target matrix is determined (826), and respective qubits of the first quantum register are encoded with tire respective phases,

[0097] The first quantum register may then be utilized while the block encoded approximation of the target matrix is operating on the first quantum register. For example, the first quantum register may be provided to a subsequent step in a quantum computation. As one example, in some embodiments the target matrix may be used to operate on or interact with another quantum register, as part of a broader quantum computation. In some embodiments, the first quantum register may be partially or entirely measured to obtain classical measurement results indicating the target matrix or aspects of the target matrix. The classical measurement results may be stored in a non- transitory computer-readable memory medium.Additional Technical Detail

[0098] The following numbered paragraphs provide additional technical detail, mathematical derivations, and description regarding embodiments herein.Definitions of Subroutines099] The following paragraphs define subroutines that are used according to some embodiments herein. Some embodiments use subroutines PREPggiven a function g, If given a functionCornp and USP, which are defined below.

[0100] Definition 2.1 PREPg: The unitary that prepares the reference state

[0101] Let g: [N] → ℝ+be a nonnegative function. Then PREPg is defined to be a unitary that generates the slate “~Sn=i g(x)\x) from an initial product state 0, i.e..

[0102] PREPgiO) -- (5)

[0103] In some embodiments, the reference function g is chosen to be computationally easy to prepare, hence PREPgis considerably easier compared to PREP for a generic target function f. Next we define the standard subroutine Ufa unitary that coherently computes the value of a given function f to an anciha.

[0104] Definition 2.2 (l / ^: Computing function values to ancilla)7246-02201PsiQ-563W01

[0105] Let f: [1, JV] R be a given function. Then If is defined to be a unitary whose action is defined as Uf|x} y) ” ix)iy + f(x)). Calculating the function values f with b-bit precision may lead to an approximation of the function values up to an additive error 2~b. Hence the number of ancilla qubits that is utilized for computing the function values with additive error e is equal to banc= max{⌈log(maxxf(x) − minxf(x))⌉, 1} + ⌈log(1 / ε)⌉, 0106] Next we introduce tire subroutine Comp that compares the values (or a function of them) registered in two of its inputs, and coherently flags the result of the comparison with 0 or 1 to a new ancilla.

[0107] Definition 2.3 (Comp Comparator)

[0108] Let f [1, / V] i-* IK+and g: [1, N] >-» IRC be given functions. Let C: (f (x), g(x),rn) {0,1} be a binary-valued function that checks ayes / no condition on f, g, and m at the point n and outputs 0 / 1, respectively for yes / no. Compcis defined to be a unitary whose action is defined as Compc| / (x)) ^(x)) m) 0) (x)) ^(x)) C( / (x), g(x), m)}. The complexity of this subroutine largely depends on the complexity of implementing the clause C.

[0109] Finally, tire subroutine USP generates a uniform state preparation over a given set of objects. It is defined as follows.

[0110] Definition 2.4 (f / SP: Uniform state preparation)

[0111] Let M G N+. Then USPMis defined to be a unitary whose action is defined as USPM|0⟩ = HEIGHT="46" WIDTH="418" SRC="imgf000021_0001.tif" / >M\C) = v M |m).

[0112] The complexity of the subroutine PREPqdepends on the function g, and the design principle is such that this subroutine is of complexity substantially less than the worst-case bound, ideally O (log / V). The choice of g together with the target function / ' determines the complexity of the other subroutines. Furthermore, note that the computed functions and the comparator may not be identical to the original functions f and g. In fact, often times, it is computationally more efficient to use other functions / ' and g in the subroutines Uyand Comp.Quantum Circuit for General State Preparation

[0113] Figure 4 illustrates a quantum circuit for the general purpose state preparation.PREPg, Uy Ug, Comp, and USP are subroutines that are used, as defined above. The reflection operator is defined as A’ i= 1 — for a given stateM G is7246-02201PsiQ-563W01chosen such that the result of the comparator leads to an approximation to the target function up to a given desired accuracy. The subroutines of tire PHASE circuit are illustrated in Figure 5. lire PHASE circuit includes the unitary operatorwhere <p: x H> <p(.r) are the phase angles. The quantum circuit shown in Figure 4 takes as input the functions f and g M 6 A+: the sampling space dimension; A: the number of amplitude amplification steps; and <p [ / V] (0,2TT): a phase angle function. The output of the quantum circuit shown in Figure 4 is an ε-approximation to the target state |ψf⟩. 'The illustrated quantum circuit may perform the following numbered, steps, in at least some embodiments.

[0114] As illustrated, the quantum registers 402-410 are initially prepared in the null state. Note that a short diagonal slash on a quantum register indicates that the register includes multiple logical qubits. The quantum register 406 does not have a slash, as this register contains a single logical qubit, the flag qubit which is utilized in conjunction with the Comp operator.

[0115] At 420, prepare the reference state \ipg) and a uniform state of dimension M: |φ1⟩ = |ψg⟩ ⊗ |USPM⟩.

[0116] At 422 and 424, compute the values f and g to ancilla: \(p2) ~ ^^ / i(hPg)|0)|0)|l7. S'PM)), which results in

[0117] \<p2) = -“2^1 ||f(x)|>|5'(%)>|m>. (6). I'ig

[0118] At 426, apply a comparator (Comp) operator to coherently flag those m for which I / (x) \M > g(x)m with 0) and the rest (those for which M / (x) < mg(x)) with Ik and obtain the state:101191 = / \ ] tnH / 'U)i>|.d(x)>~™ E;' + (7)\ s(x) / J

[0120] At 428, amplify the rmj 0 10)^ branch, and obtain

[0121] |04} = -^=2=]^ (8)Vz-zri / WI

[0122] At 430, un-compute Ug, and uniform state preparation, and obtain

[0123] At 432, add phases (optional) to the output state, and obtain | i / y ) =7246-02201PsiQ-563W01

[0124] In some embodiments, the quantum circuit shown in Figure 4 may be modified to a more general version where fy tfy, and Compcare modified to Uf, Ug, and Compcwith modified functions f, g and potentially a general clause C, respectively. This generalization may be preferable, for example, when the modified subroutines are more efficient.

[0125] Theorem 2.5 (Quantum sampling algorithm for quantum state preparation)

[0126] Let f: [1, N] → ℂ be a given complex function that can be written as f(x) = |f(x)|eiφ(x)Let f and g be the Junctions related to f and g such that the clause C is defined as

[0127] C(f(x), g(x), m) = 1 iff (9)

[0128] The circuit shown in Figure 4 prepares the state |i / rfy =and makes a number of calls to the defined subroutines, where the number of amplitude amplification steps A is given by:

[0129] (10)Design Choices for Efficient Implementation

[0130] The following paragraphs describe specific aspects of described embodiments that may improve the computational efficiency of the quantum circuit, according to some embodiments. A reference function g may be chosen such that g(x) ≥ |f(x)| for all x in the relevant domain D, and such that the reference state |ψg⟩ is efficient to prepare. For example, a constant function g(x) = c where c = maxx∈D|f(x)| is one option. The reference state for this function is particularly easy to prepare, e.g., it is the uniform state over the values of x that lies in the domain D. Depending on how the values of f fluctuate in the domain, we may want to modify the function to a piecewise constant function, such as g(x) = cvfor x e Dv, where cv= maxx∈D|f(x)|, and the subdomains Dvare nonoverlapping and UvDv= D. In these embodiments, the reference state |ψg⟩ may be generated which is a linear combination of |ψv⟩ with appropriate weights. As long as K, the number of subdomains, scales like o(N), the algorithm has the potential of having an asymptotic advantage compared to a naive QROM7246-02201PsiQ-563W01method. In some embodiments, the number of subdomains may be selected to scale as K ~ O(log N).

[0131] In some embodiments, for target states that are rapidly -decaying in some subdomains, we may employ an exponential rather than a constant function as the reference function g(x), so that the number of amplitude amplification steps is reduced. Quantum circuits that prepare these kinds of states are described in greater detail below, with examples of piece-wise constant functions (see Example 1 below), and piece-wise functions that are a mixture of constant functions and exponentials (see Example 2 below). These quantum circuits become less costly, especially when the domains and function values are chosen such that the relevant numbers are integer powers of 2.

[0132] In some embodiments, the dimension of the sampling space M, how precise the function computations are performed for computing the functions f and g to ancilla, is selectively determined to improve the efficiency and performance of the state preparation. The value of M is one of the factors that affects the efficiency of the algorithm, and it may be chosen high enough so that the comparator that effectively checks for the largest m e N+such that mg(x) < M \f (x) | is close enough to the magnitude of the function \f (x) | rescaled by some constant (that is the same for all x). Tire final desired accuracy e in preparing the target state i implies a lower bound on M, as well as an upper bound on the relative precision of computing the function values f and g to the ancilla. These results are described in greater detail below. In some embodiments, to improve efficiency in terms of the number of gates, M may be chosen to be an integer power of 2.Reference Function g

[0133] Some embodiments utilize an efficient preparation of the reference state ψg. For example, the quantum circuit PREPgmay be constructed to be computationally efficient. In some embodiments, a choice for g is selected that leads to a quantum state |ψg⟩ that may be constructed as a product state or a linear combination of a handful of product states over separate regions in the domain. In particular, g: D → ℝ may be selected as follows

[0134] g = Σ gvwhere gv: DvR, Dv∩ Dv'= ∅ for all v ≠ v', and D = ⋃vDv(18)

[0135] where each gvis a factorizable (among the bit representation of the domain) function that leads to a product state, such as a state with uniform amplitudes for all values in its7246-02201PsiQ-563W01domain. Importantly, preparing this state may utilize a quantum circuit that patches the pieces {|ψg⟩}vto the full state |ψg⟩. This may be achieved by a linear combination of unitaries and an appropriate uncompute subroutine. The full statement and the details are given in the following.A quantum circuit for PREPg

[0136] Let g = Σ gv: D → ℝ be a function that is given as a sum of K functions gv: Dv-> R where D = (JvDv, Dv∩ Dv'= ∅ for all v ≠ v'. Let PREPgbe a given quantum circuit such that PREPg|0⟩ = |ψg⟩ ∝ Σx√gv(x)|x⟩ Then there exists a quantum circuit thatuses an additional logK qubits, an additional quantum circuit of PREPαwith α =(JVgl / -^g' A'g2 / jVg> ■■■’ A^K / Ag), controlled PREPg's to create the state |ψg⟩, and a proper uncomputing circuit UN COMP if desired. In some embodiments, the UNCOMP operation may not be performed. More precisely,

[0137] |ψg⟩ = UNCOMP · (VC⊗ (PREPg)) · (PREPα⊗ ퟙ) |0⟩ ⊗ |0s⟩ (19)

[0138] where the unitary UNCOMP is given as

[0139] UNCOMP = (U ⊗ ퟙ)(∑v|v⟩⟨v| ⊗ ퟙs)(∑v,v'|v⟩⟨v'| ⊗⊗ Σx: x∈D)(|0⟩⟨0|A⊗ |x⟩⟨x|s). (20)

[0140] An example quantum circuit is shown in Figure 6 for preparing the statewhere g is given as a sum of functions gvdefined over non-overlapping domains Dv. The circuit shown in Figure 6 assumes that the description of region Dvfits into nvregisters, in particular |Dv| <Examples for Reference Function g

[0141] The following paragraphs describe explicit constructions for two examples of a reference state g. First, in Example 1, the reference function g is chosen to be piecewise constant. This function serves as a good reference function for target functions f which do not have long rapidly decaying tails / regions. A second example is described in Example 2, where the reference function g is selected to be a sum of a constant function and an exponentially decaying function. This and similar functions may serve as a good reference state for target functions f that have rapidly decaying tails or regions, such as a Gaussian. These examples are both given for functions whose domains are 1 -dimensional, i.e., x E7246-02201PsiQ-563W01[1, TV], although for some embodiments the construction may be generalized to similar functions with higher dimensional domains, with some added complexity of loading the necessary domain information to tire quantum computer.

[0142] Example 1 (Piece-wise constant) Let g = gvfor x e [NV-1, NV— 1] where No= 1 and 1 < N, < N2< ••• Nv— N, and v E [1, / f]. Namely, we wish to prepare the statefog) « SL. XXti 9v\x).

[0143] A baseline construction starts from a smaller reference state, e.g. loaded via a QROM:

[0144] (28)

[0145] Then \iq) may be obtained from 4,farse') by performing, conditioned on v), a uniform state preparation over log2( / Vv■■■■ Nv_f) qubit registers and then uncomputing the register v).

[0146] Example 2 (Exponentially decaying tails) Let g = 1 for x ∈ [0, N0− 1] and g = exp(−β|x|) for x ∈ [N0, N − 1], Namely, we wish to prepare the state |ψg⟩ ∝ Σ|x⟩ +exp(−β|x|)|x⟩.

[0147] Example 3 (Reference states for smooth target functions) Note that in several cases - including examples in this manuscript the target function f: [1, TV] -» 1R+is the N- points discretization of a smooth fimction fsmooth’. ^' -» R+, I = 1,2,3, For example, forI = 1 and fsmoothsupported in the interval [0,1], we may have f(x): = fsmooth(x / N). A natural way of defining a reference is then:

[0148] • Mesh the domain of fsmoothvia a (regular) grid with K cells {Cyji;^;

[0149] • Let gv> supx∈Cfsmooth(x) •

[0150] • Let Nvbe the number of points in the A? -point discretization that are inside Cv. In some embodiments Nvis just a constant, Nv= N / K.

[0151] • This defines a small reference state |ψgcoarse⟩ as in Eq. (28).

[0152] • Given |ψgcoarse⟩ we prepare the reference state |ψg⟩ HEIGHT="18" WIDTH="69" SRC="imgf000026_0005.tif" / >i / »5). If Nv— N / K and N / K is a power of 2, the uniform state preparations in Example 1 are simply uncontrolled products of Hadamards on the ancilla registers.

[0153] For smooth fsmooth, the convergence of the upper Riemann sums ensures that taking K = 0(1) cells is enough to ensure A = 0(1) rounds of amplitude amplification suffice in the last step of the quantum sampling algorithm. This gives a wide range of scenarios where7246-02201PsiQ-563W01quantum state preparation is asymptotically efficient if the target function can be efficiently computed to a register.Selecting the M Parameter

[0154] The choice of M is related to the accuracy of the state that is produced. The higher M is, the more accurate the state preparation is. However, that also increases the complexity of the computation. Hence, it may be desirable to pick a minimal value for M such that the resulting state is guaranteed to be approximated to an error ε, as defined above. Tire following addresses this by working out the accuracy of the comparator and how it affects the accuracy of the state preparation.M Dimension of the sampling space

[0155] Let |ψf⟩ =(1 / ‖f‖) Σxf(x)|x⟩ be tire target state, where f(x) ∈ ℝ+. and let |ψg⟩ be the reference state defined similarly to |ψg⟩ HEIGHT="18" WIDTH="16" SRC="imgf000027_0002.tif" / > where g(x) ∈ ℝ+and g(x) ≥ f(x) for all x ∈ [N]. Then, for all values of M such that

[0156] M >, (29)‖g‖1 / ‖f‖12

[0157] The quantum circuit in Figure 4 produces an e -approximation ||ψ̃f⟩ to the target state |ψf⟩, i.e., ‖ HEIGHT="18" WIDTH="71" SRC="imgf000027_0004.tif" / > |ψf⟩‖ ≤ ε. This assumes an at most e additive precision for computing the ratio f(x) / g(x), i.e., computed as f̃(x) such that

[0158] f(x) / g(x) − ε ≤ f̃(x) ≤ f(x) / g(x) + ε (30)f(x) / g(x)[01591 for 0 < d < -minrHence, a choice of ε = (1 / 4)minxg(x) leads to sufficient(1 / 4) · g(x)dimension of the sampling space

[0160] M = ⌈maxxg(x) / ε⌉ (31)Piecewise-constant g with Extended Domains

[0161] Some embodiments utilize a special case of Example 1 where the amplitude of the computational basis state x) is equal to 1 / when x ∈ [2v−1, 2v− 1]. A quantum circuit illustrating this example is shown in Figure 7, and consists of only n — 1 controlled Hadamard gates and one additional Hadamard gate. Figure 7 illustrates a quantum circuit that7246-02201PsiQ-563W01prepares a state with piecewise-constant coefficients over extended domains. More precisely, it prepares the state |ψg⟩ =

[0162] Note that the circuit is a bit more complicated (similar to the more general circuit given in Figure 6) if the boundaries of each region are not exactly an integer power of 2. More precisely, tire circuit would have up to O(K) + O(n) Toffoli gates.Exponential g: ∝ Σxexp(−βx)|x⟩[

[0163] The exponential function, exp(— / ?x), is a factorizable function. Utilizing this, in some embodiments this type of state is generated using a product of single-qubit arbitrary SU (2) rotations determined by the parameter β ∈ ℂ. More precisely, given an (n + b + l i¬ bit representation of x as x = (x−b, x−b+1, ..., x−1, x0, x1, ..., xn), i.e.,

[0164] x = Σℓxℓ· 2ℓ(45)

[0165] we express the exponential function as follows

[0166] exp(−βx) = ∏ℓ=−bne−β2x(46)

[0167] Using this expression, the target state can be expressed as a product state as follows

[0168] (1 / ‖g‖) Σxexp(−βx)|x⟩ = ⊗ℓ(1 / Nℓ) Σxe−β2x|xℓ⟩ (47)(48)

[0169] Note that the state given in Eq. (48) is a product state, each of which may be produced with a quantum circuit that implements single -qubit rotations of a given angle. Particularly, the rth state to be produced is given by

[0170] (1 / Nℓ)(|0⟩ + (49)

[0171] In some embodiments, the total cost of generating this target state may be estimated to be approximately

[0172] Cv~ (n + b + 1)(TRy+ 4 log(1 / ε))Example - Power Law Target State[

[0173] The following paragraphs describe a specific embodiment where the target state includes a power law function, e.g., it is desired to prepare |ψf⟩ ∝ Σx1 / |x|δ|x⟩ for δ > 0.7246-02201PsiQ-563W01

[0174] The set of points D = {1... 2n] may be partitioned into "annuli" indexed by g, where

[0175] = {x: 2,: 2< | | < 2fi~1}, g - 2... n + 1. (51)

[0176] The reference state g may be chosen to be piecewise constant (i.e. constant on each annulus D^y

[0177] g(x) = 2⌊log(x)⌋ + 1, (52)

[0178] The circuit to prepare g consists of two unitary operators, UVand UD. The unitary Uvloads the value of g(x) on each annulus, indexed by g. I.e. if the value of the function is given by cg, UVapplies

[0179] (53)Ar r

[0180] for some normalization constant JV\ In this case, we construct a unitary Uvthat applies

[0181] l / F|0> = 27<1“2«|g> =... O K I), (54) / .t

[0182] where the righthand equality indicates that the register indexes g are unary, This unitary can be implemented by a ladder of controlled-Ryrotations with angle θ =2 arcsin ^2 ”7!- ‘ ~2p^. Note that, in the case where β = 1, this reduces to a ladder ofcontrolled Hadamard gates. The unitary UDextends the amplitudeacross all points in the annulus, weighted by the size of the annulus:

[0183] uDcii\& ® I0)} “ = “Ezcn -y^^ |g(x),x)(55)

[0184] In the power law case, the action of UDis:

[0185] UD27(1"2« |g> ® |0)) = ® k) =® k)- (56)

[0186] Once the reference state is prepared, an ancilla register m is appended in a superposition of indices m = 0... M ■■■■ 1, and then those m for which f(x)·M > g(x)·m are coherently flagged. In this case, the inequality> x−δ·m may be checked by computing both comparands to separate registers and then carrying out a coherent inequality test.7246-02201PsiQ-563W01

[0187] Finally, the branch in which the inequality test is successful is amplified. A lower bound for the success probability of the inequality test is utilized, and therefore an upper bound on the desired number of rounds of amplitude amplification, is obtained.Example - Gaussian Target State

[0188] In some embodiments, a target state with a Gaussian profile is prepared. The target state may be described by coefficients that are proportional to a Gaussian given by

[0189] [tpf) oc exp(−x2 / σ2)|x), (67)

[0190] where, without loss of generality, we assume a Gaussian centered at x = 0, i.e., μ = 0. Our results generalize to the cases for which,u =# 0, for f(x) = exp(−x2 / σ2). We pick the domain x ∈ {−0.5, −0.5 + δ / 2, ..., −δ / 2, δ / 2, ..., 0.5 − δ / 2, 0.5} for a given 0 < 5 « 1.

[0191] The Gaussian state preparation may be handled with previous methods when σ ≪ δ and σ ≫ δ. For σ ≫ δ, the Gaussian is highly peaked at its center, hence it is more efficient to use QROM for loading the coefficients up to certain jx] < 8 + log(l / e) which implies loading only N0≪ N coefficients. In the latter case, for a > c6N for a high enough constant c < 1, a biack-box method may be used with the uniform state reference which may result in only a moderate amount of amplitude amplification round. The interesting case for which our results would lead to an improvement lie in the region, where:

[0192] δ ≪ σ ≪ δN. (68)

[0193] For these cases, using a reference function g with exponentially decaying tails is an appropriate way of reducing the number of amplitude amplification steps λ up until λ = 1. By choosing an appropriate reference state defined by g, we can minimize the number of amplitude amplification steps in the computation. The following choice, for example, implies only one round of amplitude amplification:fl, for Ixl < a,

[0194] g(x) = { exp(− (69)|x| / σ), for |x| > a.

[0195] Note that g(x) ≥ f(x) for all x. Note that with this choice, and high M, the number of amplitude amplification steps will be only λ = 1. However, this choice of g implies the following slightly modified inequality to be tested by the comparator when x > a, which is:

[0197] where M is large enough to result in at most an e approximation error in the target state. The ultimate gate count depends on the gate cost of the subroutines as well, and for7246-02201PsiQ-563W01cases 6 « o « 8N, the bottleneck is computing the Gaussian to a register for comparator (or the rotation angles via ratios of integrals of Gaussians), rather than the cost of preparing a cheaper reference state (e.g., a piecewise constant reference state) rather than a slightly more expensive one (e.g., an exponential reference state). Hence it may be desirable to minimize the number of the amplitude amplification steps.|0198J As we have seen the bottleneck is mostly in realizing the comparator given in Eq. (70), e.g., computing exp[— (|x / cr — 1 / 2)2] (or exp[— (|x| / cr — 1 / 2)2]) up to relative¬ precision e0. For a numerical resource count, we make the following choices: The function is given by / (x) = exp(— (23x)2), i.e., ff ~ 2~3, in the domain x E {—0.5, —0.5 +6,..., —6 '2, 5 / 2,..,,0.5} for a given 6 E [2"z, 2 '26] which implies an N E (10-:, 109). We also analyze the resource count for various levels of errors, in particular in the region e E [10“2, 10-9]. For these choices, A — 1 with a simple choice of piecewise constant as the reference function, such as:

[0199] g(x) = F ~2’ (7i)1j' (0.5, for \x\ > 2~~,v'

[0200] works well and implies.4 = 1.

[0201] In some embodiments, the cost of PREP(, may be estimated as only the cost of a single Y rotation with an angle of rotation that is chosen to be precise up to e / 8. This may utilize take 10 + 4log(8 / e) T-gates. Ugconsists of a comparator that coherently flags whether |x| < 2~3or | [ > 2~3, this takes 2 Toffoli gates considering the comparator.Hence, the T-cost of Ugis given by CT(Ug) = 2 for the choice of g given as in Eq. (71). The bottleneck is the cost of b. Below, we propose a comparator which we believe is more efficient than using the standard comparator that tests mg(x) < Mf(x), i.e., we modify the comparator to testing the following inequality:

[0202] lnm + (|x| / (r)2< InM, for 2“3, (72)

[0203] In m + (|x| / cr)2— In 2 < in M, for |x| > 2”3. (73)

[0204] Note that an e / 2 relative-error in exp(— x2 / a2) implies an 7 = ln(l + e / 2) « e / 2 absolute error in computing |X2 / CT2, as well as an « / 2 absolute error in computing In m. Together, these errors result in an e-relative-error of computing the amplitudes f (x). hence e0-relative error of amplitudes / '(x), which implies a total error off in approximating the target state. Computing RHS of Eq. (72) and Eq. (73) are of no considerable cost given that it is a predetermined classical value. Computing x2 / <?2exactly has Toffoli cost (log(2 / 6‘))2,7246-02201PsiQ-563W01while computing it with ri absolute-error has Toffoli cost mm{(log(2 / <5))2, (log(2 / e'))2}. For computing Inm we first express m as m = 2m°(l — in) where in e [-1 / 24 / 2] and then use the Taylor series expansion ln(l — m) =mk / k. K — log(2 / e') is sufficient to achieve an additive error of e’ / 2. Computing this series expansion utilizes K addition of powers of m, each of which hence are computed with absolute (log(X'M / 2))-bit-precision. In particular, higher powers of m are obtained iteratively via mk— mk~1m, hence each of the K multiplications costs at most (log(7< Af / 2))log(M / 2). Finally the comparator, compares the LHS with an log(lnM)-bit classical number, which is far less costly compared to the cost of the computations we just made. Hence the total cost of l / , comparator, and U is expected to be less than

[0205] 2 [(min{log(2 / 5), log(2 / e')})2+ / <log( f / 2)log(M / 2)] (74)

[0206] where K — log(2 / e'), and making the choice that e = e / 5, we have K — log(10 / e), where e: is the final error to the target state.Quantum Rejection Sampling for Matrix Block Encoding

[0207] In some embodiments, the quantum rejection sampling methods described in reference to quantum state preparation may be extended to the problem of block-encoding a matrix in a quantum register. For simplicity, some described embodiments relate to approximating a square matrix A with N x N entries. However, the described methods may be generalized to rectangular matrices as well with N x M entries, where N = M.

[0208] Some of the subroutines used for matrix block encoding are the same as described above and used for quantum state preparation. Some embodiments additionally utilize a SWAP gate and a PHASE gate acting on two quantum registers (one on a system register holding the column index, and one on the copy of the system register holding the row index of the matrix). Furthermore, instead of having one reference state preparation, matrix block encoding utilizes subroutines for two reference states, χ and φ). Various embodiments may select from a family of reference states for both % and, which may effectively lead to a reference matrix G whose elements bound the elements of the target matrix A.

[0209] Definition 5.1 (PEEffi andand their relation to matrix (?) Let A be a given target matrix to be block -encoded, and let G be a given matrix with nonnegative entries Gjj such that Gjj > ] | for alli, j G Let X' [L v] x [1, N] x [1, N] — > E with (k, i, j) ykij7246-02201PsiQ-563W01and 4>: [1, v] x [1, N] x [1, N] R with (k, i, j) « 4*^, be real-valued non-negative functions, such that

[0210] (83)

[0211] for all i,j E [1JV], where CNX'. = £f: feandJV^. = ]jTfePREPXand PREP are defined as the unitary circuits|0212] PREPx\Q}\Q}\f) = SU Xiaj VWm (84)

[0213] PREP,p\Q)\Q)\j) - (85)

[0214] where the first, second, and third registers are of v, N„ and N dimensionality, respectively.

[0215] The subroutines SWAP and PHASE are defined as follows.

[0216] Definition 5.2 (SWAP and PHASE) Let EC = (CN(CN. Then SWAP: EC C is defined as SWAP: p)[ / ) >-> | / )|i) tor all i,j G [1, N].

[0217] Furthermore, let cp: [1, N] x [1, N] -» R, then PHASE^ is defined asPHASE^: ei<ph’h |i)| / ) for all l,j £ [1, N],Figure 8 - Quantum Circuit for Matrix Block-Encoding

[0218] Figure 8 is a quantum circuit diagram illustrating an example method for performing matrix block encoding, according to some embodiments. The quantum circuit shown in Figure 8 utilizes a plurality of quantum registers 816b, 816c, 818, 820, 822 and 824 that are initially prepared in the null state. In addition, the quantum register 816a is input to the circuit prepared in a state F) which, for example, may describe a quantum system.

[0219] Classical inputs to the method may include the functions X- [ftV1x[ft N]x[ft (V] ■■■», <p'. [1, v] X [1, iV] X [1, A'] ]R, M G N+, and a phase angle function [1, A'] [0, 2TT). M may be chosen such that an at most e approximation in operator norm to the target rescaled matrix A / a is obtained. Hie quantum output of the method is the action of an o approximation in operator norm of the rescaled matrix A / a on the system register, where a = ECxJEtp. Said another way, after applying the illustrated sequence of operations, the quantum register 816a is prepared in a state 828 that results from applying an approximation of the target matrix A to the statewith a normalization factor a defined by Eq. (83), and the remaining quantum registers are returned to the null state.7246-02201PsiQ-563W01

[0220] At 802, the system state I ) =is operated on by PREPXand USPMwhich results in:|0221] I0-. ) = Sj 3=1 y~~~Xkij^Pj l?«>|O)|O)| / c)|j)| / ). (86)

[0222] At 804 and 806, respectively, the values G and A are computed to ancilla qubits by acting with UGand l / 4, to obtain:

[0223] |02O - Xf ■j' EL1 A y. J J y, IV} J J (87)

[0224] At 808, those m for which Mi / ly | > Gym are coherently flagged with |0) and the rest (those for which MIA^I < G^m) are coherently flagged with |1), wA PHASE = St, / el<p(w) | i){i | ® is applied, to obtain the state:

[0225] |03) - |m)HA(7|)|G0-)| / c)|i)| / > + 10! - X■'j(88)■F

[0226] At 810 and 812, the values ofand Gj are uncomputed in the registers, using 1.0 and U L,1T, ■ to obtain

[0227] + |1)|...). (89)

[0228] At 814, the last two registers are swapped by applying SWAP ~ Stj-i S. / XQ I |0 / | to obtain

[0229] |05) = XktpPjlty!m)!0H°)l^)! / )!0 + 11> I )• 190)

[0230] At 816, USPMand PREP^ are applied to obtain

[0231] |06) -JSU Jv (K. jrxt j |0>i0>i0>i0>i0> |0> ii> + U>i0!)|0)|0>|... > (91)

[0232] - If El!0>i0> |0> |0>|0>| 0>|i> + 11 ) 10 * >: 0>: 0>: (92)

[0233] - |0>|0>|0>|0>|0>|0>|^ + 11 )|01):0):0): (93)

[0234] Theorem 5.3 (Quantum sampling algorithm for block-encoding)7246-02201PsiQ-563W01

[0235] Let A: [1, A1] X [1, JV] « € be a given complex function that can be written as = wjgh real-valued function <p. Let x, <p: [N K] X [1,1V] X [1, N] ■■■> IRana G: [1, A] X [1,7V] -* R+be the functions such that Gy > \Ajj\for all i,j G [IV], and

[0236] V!(;(94)

[0237] Then the method described in steps 802-816 above block -encodes the matrix A / a with a rescaling factor a in the |0) branch of the ancilla states where Lf. =\Xiuj P andFigure 9 - Matrix Ziggurat - Submatrix partitioning

[0238] In some embodiments, matrix block-encoding is performed using a matrix ziggurat technique that implements submatrix partitioning. Figure 9 illustrates a quantum circuit diagram for performing matrix block encoding using submatrix partitioning, according to some embodiments. In these embodiments, the matrix A is partitioned into submatrices, where each submatrix is labeled by k G [1, v], and is a submatrix of the original matrix A. An example of submatrix partitioning for a 5x5 matrix is illustrated in Figure 10. There is a collection of lists, {5),}^, of 2 -tuples (i, j) G [1, ’] x [1, A;] that lists the row and column indices (i, / ) of the original matrix which belongs to the submatrix labeled by k, i.e., if the row-column index (i,j) belongs to the submatrix labeled by k, then that (if) appears only in the list Sk. It may be further assumed that the matrix elements of each submatrix labeled by k is of absolute value at most gk, i.e., for a given k:

[0239] < gkfor all (i,j) G Sk. (97)

[0240] For fixed / , let — {1^set°frowindices I such that (i, j) G Sk.Similarly, for fixed i let = { / ly be the set of column indexes c such that (i,j) G Sk. Note that dkj = \l^ and d'ki~ and dk= maxjdkj, and d'k= max.jd'ki. Define statesJd'kdrk)1 / 2£ik

[0241] °C SU & (98), \(diodli^^g.s

[0242] ) a yj>;X / ;y-. (99)7246-02201PsiQ-563W01

[0243] that replaces the PRE if (that prepares |^f) controlled on f) and PREP (that prepares | <ptcontrolled on i), respectively, in the quantum circuit shown in Figure 8. As a result, we obtain the block-encoding of matrix A, with the rescaling factor

[0244] a = SLi (100)

[0245] Corollary 5.5 (A block-encoding by submatrix-partitioning - Matrix ziggurat)

[0246] Let. A be a given matrix -with complex entries Ay = |z4;. ■wjprlreal-valuedfunction tp. Let {5k}f=3be a collection of lists of2-tuples (i,j) G [1, / V] X [1, / V] that lists the row and column indices (i, f) of the original matrix which belongs to the submatrix labeled by k, of dimension dkX dk, and assume gkare given such that

[0247] [riyl < gkfor all (i,f) E Sk. (101)

[0248] Furthermore, assume access to the set of row indices 1 J-^ in the submatrix k. for a given column index / , and to the set of column indices’ in the submatrix k for a given row index I. Then, a modification of the quantum circuit shown in Figure 8, where the modified circuit is illustrated in Figure 9, gives a block encoding of the matrix A with a rescaling factor a, namely, acting on |0)| T), the algorithm outputs the state

[0249] iQ)| [¥> + [0-!>|... ) (102)

[0250] where

[0251] a = ^=1 f '^kdk- (103)Figure 11 -- Row-column Block Encoding

[0252] In some embodiments, matrix block-encoding is performed using row-column block encoding. Figure 11 illustrates a quantum circuit for performing matrix block encoding using row-column block encoding, which is a modification of tire quantum circuit shown in Figure 9, according to some embodiments,

[0253] The quantum circuit shown in Figure 11 operates when Xj and <pj are selected to be of the form:

[0256] where the norms are defined as7246-02201PsiQ-563W01

[0257] || G,. ||;- Kk / k II G.:jiit- V^v, (106)

[0258] || G ||p - max || G,. Ih, || G 11^: - max || G.• Up (107)

[0259] These states define PREP (that prepares j controlled on f) and PREP (that prepares (p controlled on i), respectively.

[0260] As a result, we obtain the block-encoding of matrix A, with the rescaling factor

[0261] a = 0 Ih ll G IU (108)

[0262] where G [^ and II G 11^ are given as in Eq. (110).

[0263] Corollary 5.6 (Row-column block encoding) Let A be a given matrix with complex entries Ai}- = with real-valued function <p, and G be a matrix such that | < Ga for all i, i. Furthermore, let

[0264] || Gc. II. - Yy \ \G:!\, || G,;- ||:t-(IM (109)

[0265] || G Up — max || G;. ||p || G ||co: — max || G.,■ [[p (110)i " j

[0266] be the relevant norms related to the matrix G, and its row' and column matrix elements, and assume access to these numbers for each column index j or row index i, when specified. Then, a modification of Algorithm 5.2, given as in Fig. 11, gives a block encoding of the matrix A with a rescaling factor a, namely, acting on O'?, the algorithm outputs the state

[0267] |O)| kF> + |0L>| (Ill)

[0268] where

[0269] a - 7IG1JI G”||^. (112)Figure 12 - Column Block-Encoding

[0270] Figure 12 illustrates a quantum circuit that is another variant of the circuit shown in Figure 9 that utilizes column block-encoding, according to some embodiments.

[0271] The quantum circuit shown in Figure 12 operates when Xj and are selected to be of the form:

[0272] (H3)

[0273] |0) II G. JH / }, (114)

[0274] where7246-02201PsiQ-563W01

[0275] ]] G,;||= |GU|2.. ]] G ||F= Fl,, fiF. (115)

[0276] ’These states define PREPy(that prepares Xj controlled on / ) and PREP^ (that prepares p independent of i), respectively.

[0277] As a result, the block -encoding of matrix / I is obtained with the rescaling factor

[0278] a G ||F, (116)

[0279] where II G jjFis given as in Eq. (115).

[0280] Corollary 5.7 (Column block-encoding) Let A he a given matrix with complex entries Ati= |A.;;- |e^0- A with real-valued function <p, and G be a matrix such that \Ajj | < G^ for all l,j. Furthermore, let

[0281] || G\,. ||= |G’iyp, || G ||F= |G0|2(117)■y y

[0282] be the relevant norms related to the matrix G, and its column matrix elements, and assume access to these numbers for each column index / , when specified. Then, a modification of the method shown m Figure 9, as illustrate in Figure 12, gives a block encoding of the matrix A with a rescaling factor a, namely, acting on 0) ), the algorithm outputs the state

[0283] |0)~ |lP} + |0±>|... } (118)

[0284] where

[0285] a =|| G ||F(119)Figures 16-17 - Quantum Circuits to Emulate Time Evolution

[0286] In some embodiments, time evolution of a quantum state based on a Hamiltonian is emulated using a plurality of qubitization operators W^, as shown in Figure 16. In some embodiments, a quantum register prepared in a target quantum state may be evolved in time according to the circuit shown in Figure 16 to simulate a self-thermalization process for the target quantum state. Additionally or alternatively, time evolution may be emulated on the target quantum state as part of a simulation of a chemical reaction or other physical process. |0287] Hie time evolution of the quantum state ip) may be modelled by operating a sequence of qubitization operatorson the quantum register. Qubitization is a framework for developing a variety of types of quantum computational methods. For embodiments described herein, qubitization is employed in the context of emulating time evolution after7246-02201PsiQ-563W01preparing a quantum register in a target state. With qubitization (or more accurately, quantum signal processing), a time evolution operator may be written as a sequence of operatorsas shown in Figure 16, where the time evolution of the state ElHtas shown in 1602 is decomposed as a sequence ofoperators as shown in 1604.

[0288] The circuit shown in Figure 16 time-evolves the state 1 ) with a Hamiltonian H — (Xj Ui, which is a linear combination of unitary operatorswith real coefficients> 0. In addition, it makes use of a state £) = A-1] =i y' h where the real number A = oi-i normalizes the state. Time evolution with time t and target error e may be performed using a sequence of O(A • t + log(l / e)) qubitization operators, where the phases (i.e., angles) depend on the Hamiltonian (and time) and may be computed beforehand. The total number of phases cpj in a particular computation generally scales linearly with both the duration of time t in the time evolution, and it also increases as the energy cutoff is increased.

[0289] The decomposition of each qubitization operator W!pinto SELECT, prepare inverse (PREPT), energy cutoff (E^), and prepare (PREP) subroutines is shown in Figure 17. The SELECT subroutine applies the unitary operatorsto the 0) register (or "system register") controlled on the |L) register (or "control register"), where SELECT =Xi IO(*1|£> ® (LL)^. PREP is a subroutine that prepares the [£) state, where PREP10)®n£= |£). lire R operator 1706 applies a Z-rotation 1708 of magnitude <p when each of the input qubits tois zero.

[0290] To summarize, implementing time evolution via qubitization, involves implementing the qubitization operators 14^,, and then repeating it 0(A) times for the full computation.Additional Embodiments

[0291] The following numbered paragraphs describe additional embodiments.

[0292] In some embodiments, a quantum computational method includes preparing a first quantum register in an initial state. A preparation operator may be applied to the first quantum register to block encode a reference matrix to operate on the first quantum register. In some embodiments, the reference matrix is an approximation of a target matrix, and magnitudes of entries of the reference matrix are greater than or equal to magnitudes of entries of the target matrix for corresponding indices.

[0293] In some embodiments, rejection sampling is performed on the first quantum register to block encode an approximation of the target matrix to operate on the first quantum register. Performing rejection sampling includes rejecting samples from the first quantum register with7246-02201PsiQ-563W01values for entries of the reference matrix that exceed corresponding values of the target matrix.

[0294] In some embodiments, the reference matrix includes a plurality7of groups, where each group includes one or more cells with a respective constant value.

[0295] In some embodiments, the preparation operator is a unitary operator that outputs a summation over square roots of entries of each row or column of the reference matrix.

[0296] In some embodiments, the preparation operator is a unitary operator that outputs a Frobenius norm of the reference matrix for each row or column of the reference matrix.

[0297] In some embodiments, each entry of the reference matrix is a product of a first vector and a second vector.

[0298] In some embodiments, the reference matrix is a summation over a product of a first tensor, and a second tensor,and applying the preparation operatorto the first quantum register includes applying a PREPp operator and a PREP^p operator, or applying a PREP operator and a PREP<p operator.

[0299] In some embodiments, performing rejection sampling includes applying a first unitary operator to encode entries of the reference matrix into a second quantum register, applying a second unitary operator to encode entries of the target matrix into a third quantum register, and applying a comparator operator to entangle the second and third quantum registers with a flag register qubit. Applying the comparator operator to entangle the second and third quantum registers flags samples from the third quantum register with magnitudes of the entries of the reference matrix that are less than or equal to a maximum magnitude of the entries of the target matrix in the second quantum register for corresponding indices.

[0300] In some embodiments, after applying the comparator operator, an inverse of the first unitary operator and an inverse of the second unitary operator are applied to return the second and third quantum registers to the null state.

[0301] In some embodiments, the method further includes determining respective phases for entries of the target matrix, and encoding respective qubits of the first quantum register with the respective phases.

[0302] In some embodiments, the method further includes utilizing the first quantum register while the approximation of the target matrix is block encoded to operate on the first quantum register in a quantum computational method.7246-02201PsiQ-563W01i 0303 In some embodiments, the method further includes measuring at least a portion of the first quantum register while the approximation of the target matrix is block encoded to operate on the first quantum register to produce classical measurement results. The classical measurement results may be stored in a non-transitory computer-readable memory medium.

[0304] In some embodiments, the initial state is a state representative of a quantum system, and block encoding an approximation of tire target matrix to operate on the first quantum register applies the approximation of the target matrix to the state representative of the quantum system.

[0305] In some embodiments, a preparation operator is applied to a first quantum register and a second quantum register to prepare the first quantum register in a reference state and to prepare the second quantum register in a uniform superposition register state. The reference state is a first summation over a set of computational basis states weighted by a reference function. The reference function is an approximation of an amplitude function of a target quantum state. The target quantum state is a second summation over the computational basis states weighted by the amplitude function, and a magni tude of the reference function is greater than or equal to a magnitude of the amplitude function for corresponding arguments.

[0306] In some embodiments, rejection sampling is performed on the first quantum register to prepare the first quantum register in the target quantum state. Performing rejection sampling includes rejecting samples from the second quantum register based at least in part on a comparison of a magnitude of the reference function and a magnitude of the amplitude function.

[0307] In some embodiments, a preparation operator is applied to a first quantum register and a second quantum register to prepare the first quantum register in a reference state and to prepare the second quantum register in a uniform superposition register state. The reference state is a first summation over a set of computational basis states, x), weighted by a reference function,tq(x). The reference function is an approximation of an amplitude function, / '(x), of a target quantum state. The target quantum state is a second summation over the computational basis states weighted by the amplitude function. A magnitude of the reference function is greater than or equal to a magnitude of the amplitude function for corresponding values of x.

[0308] In some embodiments, a first unitary operator is applied to encode the amplitude function of the target quantum state into a third quantum register.7246-02201PsiQ-563W01

[0309] In some embodiments, a second unitary operator is applied to encode the reference function of the reference state into a fourth quantum register.

[0310] In some embodiments, a comparator operator is applied to entangle the second, third and fourth quantum registers with a flag register qubit. Applying the comparator operator to entangle the second, third and fourth quantum registers flags samples from the fourth quantum register with magnitudes of the reference function that are less than or equal to a maximum magnitude of the amplitude function in the third quantum register for corresponding computational basis states.

[0311] In some embodiments, an amplitude amplification subroutine is iteratively applied to amplify the flagged samples from the fourth quantum register.

[0312] In some embodiments, an inverse of the preparation operator, an inverse of the comparator operator, and an inverse of the first unitary operator are applied to return the second and third quantum registers to a null state and to prepare the first quantum register in an approximation of the target quantum state.

[0313] In some embodiments, the method further includes determining respective phases for a plurality of eigenstates of the target quantum state; and encoding respective qubits of the first quantum register with the respective phases.

[0314] In some embodiments, a first quantum register is prepared in an initial state, a PREPXoperator is applied to the first quantum register, a USPM operator is applied to a second quantum register to prepare the second quantum register in a uniform superposition register state, and a UG unitary operator is applied to the first quantum register and a third quantum register to encode entries of a reference matrix into the third quantum register. The reference matrix is an approximation of a target matrix, and magnitudes of entries of the reference matrix are greater than or equal to magnitudes of entries of the target matrix for corresponding indices.

[0315] In some embodiments, a UA unitary operator is applied to the first quantum register and a fourth quantum register to encode entries of the target matrix in the fourth quantum register.

[0316] In some embodiments, a comparator operator is applied to entangle the second, third and fourth quantum registers with a flag register qubit. Applying the comparator operator to entangle the second, third and fourth quantum registers with the flag register qubit flags samples from the third quantum register with magnitudes of entries of the reference matrix7246-02201PsiQ-563W01that are less than or equal to a magnitude of entries of the target matrix in the third quantum register for corresponding indices.

[0317] In some embodiments, an inverse of the UG unitary operator is applied to the first and third quantum registers to return the third quantum register to a null state, an inverse of the UA unitary operator is applied to the first and fourth quantum registers to return the fourth quantum register to the null state, an inverse of the USPM operator is applied to the second quantum register to return the second quantum register to the null state, and an inverse of a PREP operator is applied to the first quantum register to block encode an approximation of the target matrix to operate on the first quantum register.

[0318] It should be understood that all numerical values used herein are tor purposes of illustration and may be varied. In some instances, ranges are specified to provide a sense of scale, but numerical values outside a disclosed range are not precluded.

[0319] It should also be understood that all diagrams herein are intended as schematic.Unless specifically indicated otherwise, the drawings are not intended to imply any particular physical arrangement of the elements shown therein, or that all elements shown are necessary'. Those skilled in the art with access to this disclosure will understand that elements shown in drawings or otherwise described in this disclosure may be modified or omitted and that other elements not shown or described may be added.

[0320] This disclosure provides a description of the claimed invention with reference to specific embodiments. Those skilled in the art with access to this disclosure -will appreciate that the embodiments are not exhaustive of the scope of the claimed invention, which extends to all variations, modifications, and equivalents.

[0321] The terminology used in the description of the various described embodiments herein is for the purpose of describing particular embodiments only and is not intended to be limiting. As used in the description of the various described embodiments and the appended claims, the singular forms “a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will also be understood that the term “and / or” as used herein refers to and encompasses any and all possible combinations of one or more of the associated listed items. It will be further understood that the terms “includes.” “including,” “comprises,” and / or “comprising,” when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and / or components, but do7246-02201PsiQ-563W01not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof.

[0322] It will also be understood that, although the terms first, second, etc., are, in some instances, used herein to describe various elements, these elements should not be limited by these terms. These terms are only used to distinguish one element from another. For example, a first switch could be termed a second switch, and, similarly, a second switch could be termed a first switch, without departing from the scope of the various described embodiments. The first switch and the second switch are both switches, but they are not the same switch unless explicitly stated as such.

[0323] As used herein, the term “if’ is, optionally, construed to mean “when” or “upon” or “in response to determining” or “in response to detecting” or “in accordance wi th a determination that,” depending on the context.

[0324] ’Die foregoing description, for purpose of explanation, has been described with reference to specific embodiments. However, the illustrative discussions above are not intended to be exhaustive or to limit the scope of the claims to the precise forms disclosed. Many modifications and variations are possible in view of the above teachings. The embodiments were chosen in order to best explain the principles underlying the claims and their practical applications, to thereby enable others skilled in the art to best use the embodiments with various modifications as are suited to the particular uses contemplated.

Claims

7246-02201PsiQ-563W01ClaimsWhat is claimed is:

1. A method, comprising:applying a preparation operator to a first quantum register and a second quantum register to prepare the first quantum register in a reference state and to prepare the second quantum register in a uniform superposition register state, wherein the reference state comprises a first summation over a set of computational basis states, |x), weighted by a reference function, g(x'), wherein the reference function comprises an approximation of an amplitude function,of a target quantum state, wherein the target quantum state composes a second summation over the computational basis states weighted by the amplitude function, and wherein a magnitude of the reference function is greater than or equal to a magnitude of the amplitude function for corresponding values ofx; andperforming rejection sampling on the first quantum register to prepare the first quantum register in the target quantum state, wherein performing rejection sampling comprises rejecting samples from the second quantum register based at least in part on a comparison of a magnitude of the reference function,,g(x) i, and a magnitude of the amplitude function, | / (x) |.

2. The method of claim 1,wherein perfonning rejection sampling comprises:applying a first unitary operator to encode the amplitude function of the target quantum state into a third quantum register;applying a second unitary operator to encode the reference function of the reference state into a fourth quantum register;applying a comparator operator to entangle the second, third and fourth quantum registers with a flag register qubit, wherein applying the comparator operator to entangle the second, third and fourth quantum registers flags samples from the fourth quantum register composing magnitudes of the reference function that are less than or equal to a maximum magnitude of the amplitude function in the third quantum register for corresponding computational basis states; andapplying an amplitude amplification operation to amplify the flagged samples from the fourth quantum register.7246-02201PsiQ-563W013. lire method of claim 2, further comprising:after applying the amplitude amplification operation:applying an inverse of the preparation operator, an inverse of the comparator operator, and an inverse of the first unitary operator to return the second and third quantum registers to a null state.

4. The method of claim 2, further comprising:after applying the amplitude amplification operation:determining respective phases for a plurality of eigenstates of the target quantum state; andencoding respective qubits of the first quantum register with the respective phases.

5. The method of claim 1,wherein the reference function comprises one or more of:a constant or piecewise constant function;a linear or piecewise linear function; andan exponential or piecewise exponential function.

6. The method of claim 5,wherein the piecewise linear function shares values with one or more maxima of the amplitude function.

7. The method of claim 5,wherein the exponential function tracks a Gaussian profile of the amplitude function.

8. The method of claim 5, further comprising:determining the piecewise linear function by:dividing a domain of the amplitude function into plurality of subdomains; and within each subdomain, identifying a respective magnitude of the piecewise linear function as a maximum magnitude of the amplitude function within the respective subdomain.7246-02201PsiQ-563W019. lire method of claim 1, further comprising:prior to applying the preparation operator to the first quantum register and the second quantum register, preparing the first and second quantum registers in a null state.

10. The method of claim 1, further comprising:utilizing the first quantum register prepared in the target quantum state to prepare a purified thermal state;utilizing the first quantum register prepared in the target quantum state to prepare an initial state of a quantum computation; orutilizing a quantum circuit that prepares the target quantum state as a subroutine for matrix block-encoding.

11. The method of claim 10, further comprising:utilizing the purified thermal state to perform a quantum simulation of a chemical reaction.

12. A quantum computing system, comprising:a non-transitory computer-readable memory medium;a plurality of physical qubits; anda controller, wherein the quantum computing system is configured to:apply a preparation operator to a first quantum register and a second quantum register to prepare the first quantum register in a reference state and to prepare the second quantum register in a uniform superposition register state, wherein the reference state comprises a first summation over a set of computational basis states weighted by a reference function, wherein the reference function comprises an approximation of an amplitude function of a target quantum state, wherein the target quantum state comprises a second summation over the computational basis states weighted by the amplitude function, and wherein a magnitude of the reference function is greater than or equal to a magnitude of the amplitude function for corresponding arguments;perform rejection sampling on the first quantum register to prepare the first quantum register in the target quantum state, wherein performing rejection sampling comprises rejecting samples from the second quantum register based at least in part on a7246-02201PsiQ-563W01comparison of a magnitude of the reference function and a magnitude of the amplitude function.

13. The quantum computing system of claim 12,wherein, in performing rejection sampling on the first quantum register, the quantum computing system is configured to:apply a first unitary operator to encode the amplitude function of the target quantum state into a third quantum register;apply a second unitary operator to encode the reference function of the reference state into a fourth quantum register;apply a comparator operator to entangle the second, third and fourth quantum registers with a flag register qubit, wherein applying the comparator operator to entangle the second, third and fourth quantum registers flags samples from the fourth quantum register comprising magnitudes of the reference function that are less than or equal to a maximum magnitude of the amplitude function in the third quantum register tor corresponding computational basis states; anditeratively apply an amplitude amplification subroutine to amplify the flagged samples from the fourth quantum register.

14. The quantum computing system of claim 12,wherein the quantum computing system is further configured to:apply an inverse of the preparation operator, an inverse of the comparator operator, and an inverse of the first unitary operator to return the second and third quantum registers to a null state and to prepare the first quantum register in an approximation of the target quantum state.

15. The quantum computing system of claim 12,wherein the quantum computing system is further configured to:determine respective phases for a plurality of eigenstates of the target quantum state; andencode respective qubits of the first quantum register with the respective phases.7246-02201PsiQ-563W0116. The quantum computing system of claim 12,wherein the quantum computing system is further configured to:utilize the first quantum register prepared in the target quantum state to prepare a purified thermal state;utilize the first quantum register prepared in the target quantum state to prepare an initial state of a quantum computation; orutilize a quantum circuit that prepares the target quantum state as a subroutine for matrix block-encoding.

17. The quantum computing system of claim 12,wherein the quantum computing system is further configured to:utilize the purified thermal state to perform a quantum simulation of a chemical reaction.

18. A non-transitory computer-readable memory medium storing program instructions which, when executed by a processor, cause a quantum computing system to:prepare a first quantum register in an initial state;apply a PREP operator to the first quantum register;apply a USPM operator to a second quantum register to prepare the second quantum register in a uniform superposition register state;apply a UGunitary operator to the first quantum register and a third quantum register to encode entries of a reference matrix into the third quantum register, wherein the reference matrix comprises an approximation of a target matrix, and wherein magnitudes of entries of the reference matrix are greater than or equal to magnitudes of entries of the target matrix for corresponding indices,apply a UA unitary operator to the first quantum register and a fourth quantum register to encode entries of the target matrix in the fourth quantum register;apply a comparator operator to entangle the second, third and fourth quantum registers with a flag register qubit, wherein applying the comparator operator to entangle the second, third and fourth quantum registers with the flag register qubit flags samples from the third quantum register comprising magnitudes of entries of the reference matrix that are less7246-02201PsiQ-563W01than or equal to a magnitude of entries of the target matrix in the third quantum register for corresponding indices;apply an inverse of the UG unitary operator to the first and third quantum registers to return the third quantum register to a null state;apply an inverse of the UA unitary operator to the first and fourth quantum registers to return tire fourth quantum register to the null state;apply an inverse of the USPM operator to the second quantum register to return the second quantum register to the null state; andapply an inverse of a PREPXoperator to the first quantum register to block encode an approximation of the target matrix to operate on the first quantum register.

19. The non-transitory computer-readable memory medium of claim 18, wherein the program instructions are further executable to cause the quantum computing system to:utilize the first quantum register prepared in the target quantum state to prepare a purified thermal state;utilize the first quantum register prepared in the target quantum state to prepare an initial state of a quantum computation; orutilize a quantum circuit that prepares the target quantum state as a subroutine for matrix block -encoding.

20. The non-transitory computer-readable memory medium of claim 19, wherein the program instructions are further executable to cause tire quantum computing system to:utilize the purified thermal state to perform a quantum simulation of a chemical reaction.

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