Quantum device initialisation

Abstract

A quantum computing system 100 comprises a compiling system 102 configured to receive a quantum state specification comprising a plurality of state variables defining a quantum state, determine a comp

Description

Field of the invention The present invention relates to quantum computation. Background Quantum computers have the potential to perform computations that would be intractable on even the most powerful classical computers. Instead of representing information using classical bits, quantum computers generally use qubits that can be in a simultaneous superposition of multiple quantum states. As quantum computing technology matures, there is an increasing acceptance that high-quality qubits alone will not be sufficient to achieve quantum advantage. For example, large-scale fault-tolerant quantum computation will require advances in areas such as quantum error correction and quantum control, which are generally handled by classical computing devices. Another area that could undermine quantum advantage is initialising quantum devices in specific quantum states. Quantum algorithms often require specific quantum states to be initialised on a quantum computer, and preparing these states can require large numbers of qubits and quantum operations. There is therefore a need for improved methods and devices for preparing quantum states on a quantum computer. Summary of the invention According to a first aspect of the invention, there is provided a quantum computing system comprising: a compiling system; and a quantum processing unit comprising a control system coupled to a plurality of quantum devices, wherein the compiling system is configured to: receive a quantum state specification comprising a plurality of state variables defining a quantum state; determine a compressed representation of the quantum state, wherein determining the compressed representation of the quantum state comprises: splitting the plurality of state variables into a plurality of state variable subsets; and determining a respective polynomial approximation representing each respective subset of the plurality of state variable subsets, and send the compressed representation to the control system, and wherein the control system is configured to: receive the compressed representation of the quantum state; and use the compressed representation to prepare the plurality of quantum devices according to the quantum state. Preparing the quantum states exactly on a quantum computer requires substantial quantum resources and risks undermining any potential quantum advantage. The present invention addresses this problem by determining a compressed representation of the quantum state that can be represented using fewer parameters than the exact representation. Existing approaches for approximating quantum states fit a single high-degree polynomial to all state variables. Compared to these approaches, by splitting the state variables into multiple subsets each with a different polynomial approximation, the present invention facilitates compression with improved accuracy (more granular polynomial fitting / approximation is achieved by focusing on subsets of state variables rather than attempting to approximate all state variables with a single polynomial) and with lower-degree polynomials (which are generally easier / faster to fit compared to higher-degree polynomials and require fewer coefficients to represent). Instead of being defined by the state variables, the compressed representation is defined by the polynomial approximations (e.g. coefficients of the polynomial approximations). The compressed representation is used to prepare the quantum devices in a state that approximates the quantum state (i.e. the target / desired quantum state defined by the state specification). The compressed representation is therefore a lossy compression of the quantum state. Due to the combination of improved approximation accuracy and lower-degree polynomials, the present invention enables a greater level of compression (i.e. requires fewer polynomial coefficients) to achieve the same degree of accuracy as existing approaches. In addition, the use of multiple polynomial approximations is more versatile and can handle discontinuities between state variables (i.e. large jumps between successive state variables). Using the compressed representation to prepare the plurality of quantum devices according to the quantum state involves using the compressed representation (e.g. the polynomial coefficients or control signals derived therefrom) to prepare the plurality of quantum devices in a state that approximates the desired quantum state (i.e. to within some predetermined accuracy - the prepared state approximates the desired quantum state due to the lossy nature of the compression). One skilled in the art will appreciate that various quantum circuits may be used to prepare quantum devices based on polynomial coefficients. The quantum devices may be any quantum devices capable of storing quantum information (i.e. any devices suitable for encoding information using quantum computational states). The quantum devices may be qubits. Alternatively, the quantum devices may be other devices capable of storing quantum information, such as qudits or qutrits. While the description herein will primarily refer to qubits, any reference herein to qubits should be understood to also encompass other types of quantum devices unless explicitly stated otherwise. A quantum computing system (also referred to herein as a quantum computer) is a computing system that exploits quantum mechanical phenomena (i.e. using quantum devices). A quantum processing unit is a device (such as a chip) with one or more quantum devices (such as qubits) that are controlled by a control system (e.g. the control system may transmit RF pulses for performing operations and readout on the quantum devices). A control system (also referred to as a quantum control system) is generally a classical processing device configured to communicate with quantum devices (e.g. by sending and receiving control signals such as RF pulses) to perform operations such as quantum logic gates and state measurement / readout. The compiling system is a classical processing system that processes a quantum state specification to generate a compressed representation of the quantum state specification for use by the control system. The state variables may be coefficients of eigenstates of the quantum state. The compressed representation may optionally be processed (e.g. by the compiling system, the control system or some other system) from a high level format (e.g. coefficients of the polynomial approximations) into a low level format (such as a series of quantum gate / quantum circuit instructions that utilise the polynomial coefficients to prepare the quantum devices according to the quantum state, e.g. instructions that are derived from / determined based upon the polynomial coefficients, or RF pulses / control signals representing such gate instructions). The compressed representation may therefore comprise polynomial coefficients of the respective polynomial approximations and / or signals or data derived from the polynomial coefficients. The compressed representation may comprise quantum circuit instructions (i.e. quantum circuit instructions encoding (or otherwise based on) the polynomial approximations). A polynomial approximation is a polynomial function that can be used to determine approximate values for the state variables. Each polynomial approximation comprises a plurality of polynomial coefficients. Each polynomial approximation has an associated degree (or degree value), which is the highest of the degrees of the polynomial’s individual terms with non-zero coefficients (for example, a cubic polynomial of the form cx + c2% + c3%2 + c4%3 has degree three and four polynomial coefficients (which, except for c4, may optionally be zero). The number of coefficients associated with each polynomial approximation is generally one larger than the degree (a degree-three cubic polynomial has four polynomial coefficients). Coefficients may be zero, and they may be represented as signed or unsigned fixed point numbers, floating point numbers or integers. The polynomials may optionally have definite parity (i.e. contain only even or only odd coefficients), in which case the number of coefficients associated with each polynomial will be roughly half the degree. Any of the various techniques known to a person skilled in the art may be used to obtain the polynomial approximations, including polynomial regression, curve fitting (e.g. least squares), polynomial interpolation, Chebyshev approximation and the Remez algorithm. The polynomial approximation may use any suitable polynomial basis such as the monomial basis or Chebyshev polynomials. Preparing the plurality of quantum devices according to the quantum state may involve transmitting control signals to the quantum devices based on the polynomial coefficients (e.g. control signals to perform quantum operations such as phase rotations based on the coefficient values). The plurality of state variables may have an associated size value, and a size of each of the plurality of state variable subsets may be equal to the size value. The size value may be a size of a data array encoded in the quantum state (e.g. a number of rows / columns) or may be some other size value, such as a size of a physical system represented by the quantum state. Many quantum states of interest have a natural grouping of values corresponding to a size value of a physical system of interest. For example, a quantum state may include values representing a two-dimensional (or higher dimensional) physical parameter, such as pressure. When represented as a two-dimensional data array, such values will generally be relatively smooth (i.e. relatively small changes between adjacent values). However, when the data is transposed into a one-dimensional quantum state (which is a vector rather than a two-dimensional matrix), large changes in value can occur between adjacent values where transitions between matrix rows occur. Finding polynomial approximations that accommodate such transitions is difficult and generally reduces the accuracy of the fitting. Splitting the state variables into subsets based on a size value (e.g. a size an original data array, such as the width / number of columns) ensures that each subset contains only smooth transitions between adjacent values, thereby providing accurate fittings with low-degree polynomials, which reduces the number of coefficients required to approximate the quantum state, thereby increasing the level of compression and reducing the resources required to prepare the quantum state. All respective polynomial approximations may optionally have equal polynomial degree. A total number of polynomial coefficients associated with all respective polynomial approximations is preferably less than a total number of state variables in the plurality of state variables. Each of the plurality of state variable subsets preferably comprises multiple state variables. Each of the plurality of state variable subsets may comprise an equal number of state variables (i.e. all respective state variable subsets have equal size). Optionally, each respective state variable subset may comprise a plurality of sequential state variables. That is, each state variable subset may comprise an ordered subsequence of the ordered sequence of all of the state variables. According to a second aspect of the invention, there is provided a method of initialising a plurality of quantum devices in a quantum computing system, the method comprising: receiving, at a compiling system of the quantum computing system, a quantum state specification comprising a plurality of state variables defining a quantum state; determining, at the compiling system, a compressed representation of the quantum state, wherein determining the compressed representation of the quantum state comprises: splitting the plurality of state variables into a plurality of state variable subsets; and determining a respective polynomial approximation representing each respective subset of the plurality of state variable subsets, sending the compressed representation from the compiling system to a control system of the quantum computing system; and using the compressed representation, at the control system, to prepare the plurality of quantum devices according to the quantum state. The second aspect of the invention provides the same benefits as the first aspect of the invention. Any feature described in combination with the first aspect of the invention may also be combined with the second aspect of the invention. According to a third aspect of the invention, there is provided a computer-readable medium (such as a non-transitory computer readable medium) comprising instructions which, when executed by a quantum computing system comprising a compiling system, a control system and a plurality of quantum devices, cause the quantum computing system to carry out the method of the second aspect of the invention. Brief description of the drawings Examples of the present invention will now be described in detail with reference to the accompanying drawings, in which: Fig. 1 is a schematic of a quantum computing system; Fig. 2 is a flowchart showing a method of preparing quantum devices according to a quantum state; and Fig. 3 shows an example of a quantum circuit for use in preparing quantum devices according to a quantum state. Detailed description A schematic of an exemplary quantum computing system 100 for performing the methods of the present disclosure is shown in Fig. 1. The quantum computing system 100 comprises a plurality of quantum devices 106, such as physical or logical qubits. The quantum devices 106 are controlled by a control system 104 (also referred to herein as a quantum control system), and both the quantum devices 106 and control system 104 are part of a quantum processing unit (QPU) 108. The control system 104 is generally a classical processing device that transmits control signals to the quantum devices 106 for performing operations on the quantum devices 106 (including measurement operations) and receives measurement information from the quantum devices 106. The measurement information will generally be analogue data signals, although the analogue signals may alternatively be converted to digital signals before being transmitted to the control system 104 in some implementations (e.g. the quantum devices 106 may be provided with one or more analogue-to-digital converters). The control system 104 may receive high-level instructions and convert these high-level instructions (such as logic gates) into low-level qubit instructions (e.g. microwave pulses etc.), which may be in analogue format. The illustrated quantum computing system 100 additionally comprises a compiling system 102. The compiling system 102 is operable to receive quantum circuit specifications and quantum state specifications for sending to the control system 104. The compiling system 102 may process the circuit and state specifications to generate updated circuit / state specifications and / or translate high-level circuit / state specifications into low-level machine code or pulse specifications that the control system 104 can use to perform operations on the quantum devices 106 (such as preparing quantum states). One skilled in the art will appreciate that the quantum computing system 100 may also comprise additional components, including intermediary components positioned between the illustrated components, and that the illustrated components may be connected in a different configuration. For example, the quantum computing system 100 may additionally comprise a decoding system used to perform quantum error correction. The quantum computing system 100 of Fig. 1 may be used to prepare the quantum devices 106 according to a quantum state by determining a compressed representation of the quantum state at the compiling system 102 and using the compressed representation at the control system 104 to prepare the quantum devices 106. A flowchart showing a method for preparing the quantum devices 106 according to a quantum state is shown in Fig. 2. In a first step 201, the compiling system 102 receives a quantum state specification comprising a plurality of state variables defining a quantum state. For example, a quantum state |tp) may be defined as N-l ivo = i=0 where yt are the state variables and |i) are quantum eigenstates (the state variables are coefficients of the eigenstates). The example quantum state |tp) involves n = log2 N qubits and requires N state variables (which may be complex). The state |tp) is essentially a superposition of A? orthogonal states, with each respective state variable yt representing the amplitude of the corresponding state |i). The state variables yt may be real or complex numbers, and the state |tp) will preferably be normalised such that (V'lV') = ytyt = 1-The compiling system 102 then determines a compressed representation of the quantum state in step 202. Preparing the state |tp) exactly on a quantum computer would generally require loading N parameters (e.g. the N coefficients yt, which may be complex numbers) into the quantum computer. Fault-tolerant implementation of such data loading requires substantial quantum resources and risks undermining any potential quantum advantage. The present invention addresses this problem by determining a compressed representation of the quantum state that can be represented using fewer parameters than the exact representation. Determining the compressed representation is performed by splitting the plurality of state variables into a plurality of state variable subsets and determining a respective polynomial approximation representing each respective subset of the plurality of state variable subsets (i.e. a different polynomial representation for each respective state variable subset). A polynomial approximation is a polynomial function that can be used to determine approximate values for the state variables. Each polynomial approximation comprises a plurality of polynomial coefficients. While the examples herein use polynomials in the monomial basis, one skilled in the art will appreciate that alternative bases, such as Chebyshev polynomials, may also be used. Each state variable subset contains multiple state variables, and the subsets are preferably of the same size (i.e. contain the same number of state variables). The state variables in each subset are preferably sequential (i.e. each subset is preferably a subsequence of the ordered sequence of all state variables) or selected from the ordered sequence of all state variables at regular intervals (e.g. an interval that corresponds to a size value associated with the state variables). In other words, the subsets may be obtained by ordering all state variables (i.e. in ascending or descending order according to the index i) and selecting ordered subsequences (e.g. sequences having a regular size that may correspond to a size value associated with the state variables) or by selecting state variables at given intervals (e.g. every jth element of the ordered sequence, where j may optionally correspond to the size value). Assuming that the state variables are split into M equal subsets each containing L state variables (such that M xL = N), can be represented as N-l L-1M-1 IVO = y; 10 = ymL+i\mL +1). i = 0 1=0 m=0 Each subset of state variables (i.e. each subset having a particular value of m) can then be processed independently to find a polynomial function that approximates the values of the state variables. For example, the first subset (m = 0) has L state variables y0, ...,yL-i. Conventional fitting techniques can be used to determine a polynomial function, p0(x) that approximates these state variables such that y( « p^l / L^O <1<L. This process can be repeated for each value of m to obtain M polynomial functions pm(x) for 0 <m <M such that ymL+i ~ Pm(l / ^> o <m <M,0 <I <L. Each polynomial function pm(x) will have an associated degree dm (i.e. the highest of the degrees of the polynomial’s individual terms with non-zero coefficients); the degree of a polynomial function determines how many coefficients are required to define the polynomial (each polynomial approximation generally requires dm + 1 coefficients to represent it rather than the L parameters that would be required to represent the state values approximated by that polynomial, although some polynomials (e.g. those with definite parity) may require fewer coefficients). To ensure that the compression procedure reduces the number of variables required to represent the quantum state, the total number of polynomial coefficients should preferably be less than the number of state variables. One way to ensure this is by selecting polynomials having equal degree d with (d +1) x M <N (or equivalently, to have d + 1 <L). Increasing the value of d will generally lead to more accurate approximation of the quantum state at the cost of requiring more resources to identify polynomials and to prepare the quantum state. Once the compressed representation has been determined, the compiling system 102 sends the compressed representation to the control system 104 in step 203, possibly via one or more intermediate devices. The compressed representation may optionally be processed (e.g. by the compiling system 102, the control system 104 or some other system) from a high level format (i.e. the coefficients of the polynomial approximations) into a low level format (such as a series of quantum gate / quantum circuit instructions that utilise the polynomial coefficients (i.e. are derived from / determined based upon the polynomial coefficients) to prepare the quantum devices according to the quantum state, or RF pulses / similar control signals representing such gate instructions). The compressed representation of the quantum state is received at the control system 104, and in step 204 the control system 104 uses the compressed representation to prepare the quantum devices according to the quantum state. In other words, the control system 104 uses the information encoded in the compressed representation (e.g. the polynomial coefficients, or quantum gates / RF signals that are determined based on the polynomial coefficients) to prepare the quantum devices in a state that approximates the desired quantum state (e.g. within a predetermined accuracy, which can be imposed when determining the compressed representation of the quantum state). Preparing the quantum devices according to the quantum state can be performed using a quantum circuit such as the quantum circuit 300 shown in Fig. 3, which shows an example of a quantum circuit that can be used to prepare the quantum devices for polynomial approximations of degree three having definite parity and real amplitudes. The quantum circuit 300 involves a quantum singular value transformation (QSVT) circuit with the block encoding U, where r U = 2 / L \ (L-1) / J Multi-controlled rotations by phase factors <pim (where i = 0,..., d-1) are performed dependent upon (i.e. controlled by) the top register (forward slashes on horizontal qubit lines denote registers of qubits). The phase factors (which can be determined using techniques such as those disclosed in arXiv:2308.01501v2 [quant-ph], which is hereby incorporated by reference) determine which polynomial is applied by the QSVT circuit, so there is a different polynomial pm(x) for each m. The flag qubits are initialised and postselected as zero. The symbol 0 on a controlled operation represents multiplexed control, i.e. where the operation depends upon the state of the qubit register (for example, if the control register is in state m, the operation Rz(<pi:m) is performed on the target qubit / register). Using the quantum circuit 300 of Fig. 3 provides a block encoding of the desired diagonal matrix: / L - 1\ p°[—) / 0\ Pm^ \                                  Pm-i (-[-) / This matrix can be applied to an equal superposition quantum state (e.g. prepared in the top and bottom registers of qubits in Fig. 3) and postselecting the flag qubits in the |0) state to prepare qubits according to the desired quantum state (postselection may necessitate multiple attempts to obtain the desired state - one skilled in the art will appreciate that various approaches can be used to increase the success probability). One skilled in the art will appreciate that various methods and quantum circuits can utilised with the compressed representation to prepare the plurality of quantum devices according to the quantum state, and the circuit of Fig. 3 is merely one example of such a quantum circuit. For example, one could instead use generalized quantum eigenvalue transformation (GQET) techniques (or any other QSVT-like algorithm) to perform polynomial transformations as determined by phase factors. In addition, while the above example uses polynomial functions that act on arguments (inputs) of the form l / L, one skilled in the art will recognise that alternative approaches may use different forms of arguments, such as block encodings of the form sin(Z / L) (i.e. with yi,m ~ Pm(sin (V^))). 'n which case a block encoding U' can be used instead of U, where / sin(0 / L) sin(l / L) U' = \ and the desired block encoding is / po(sin(O / L)) p0(sin ((L — 1) / L)) \ pM_i(sin ((L - 1) / L))> Other arguments, xh having different forms may also be used (the above examples use Xi = lfL and xv = sin(Z / L) respectively). sin(2 / L) pM_i(sin(0 / L)) Block encodings U and U' can be implemented using techniques that are common general knowledge to one skilled in the art (for example using the techniques in Watts et al., Efficient amplitude encoding of polynomial functions into quantum computers, July 2023, arXiv:2307.10917v3 [quant-ph], and McArdle et al., Quantum state preparation without coherent arithmetic, October 2022, arXiv:2210.14892v1 [quant-ph] respectively, both of which are hereby incorporated by reference). Likewise, the multi-controlled rotations in Fig. 3 can also be implemented using techniques that are common general knowledge to one skilled in the art, such as those disclosed in, arXiv:quant-ph / 0406176v5, Babbush et al., Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity, Physical ReviewX, 8(4):041015, October2018, DOI: 10.1103 / PhysRevX.8.041015, and Hao Lowet al., Trading T-gates for dirty qubits in state preparation and unitary synthesis, December 2018, arXiv:1812.00954v1 [quant-ph], all of which are hereby incorporated by reference. Existing approaches for approximating quantum states fit a single high-degree polynomial to all state variables. Compared to these approaches, the present invention facilitates improved approximation (more granular fitting / approximation can be achieved by focusing on subsets of state variables rather than attempting to approximate all state variables with a single polynomial) with lower-degree polynomials (which are generally easier / faster to fit compared to higher-degree polynomials). In addition, many quantum states of interest have a natural grouping of values corresponding to a size value of a physical system of interest. For example, a quantum state may include values representing a two-dimensional (or higher dimensional) physical parameter, such as pressure. When represented as a two-dimensional data array, such values will generally be relatively smooth (i.e. relatively small changes between adjacent values). However, when the data is transposed into a onedimensional quantum state (which is a vector rather than a two-dimensional matrix), large transitions can occur between adjacent values where transitions between matrix rows occur. Finding polynomial approximations that accommodate such transitions is difficult and generally reduces the accuracy of the fitting. However, the present invention can split the state variables into subsets based on the size value (i.e. a size the original array, such as the width / number of columns) such that each subset contains only smooth transitions between adjacent values. It should be understood that any method of the present disclosure could include additional steps, and any device could include additional components. In addition, unless indicated otherwise or technically infeasible, the method steps disclosed herein may be performed in alternative orders, and any order described herein should be considered as exemplary rather than limiting. The illustrated steps and components could be split into multiple sub-steps / subcomponents. Furthermore, one skilled in the art will appreciate that any computation that can be performed by a classical processing device can also be performed by a quantum computing device. Accordingly, any methods or described herein that is performed on a classical computing device (such as a CPU) can also be performed by a quantum processing device, 5 such as a quantum processing unit (QPU) comprising a plurality of qubits. While the above examples use qubits as the quantum devices, it should be understood that these examples could alternatively be implemented with other types of quantum devices, such as qutrits or qudits, and that the invention disclosed herein is applicable to both qubits and other types of quantum devices.

Claims

1. A quantum computing system comprising:a compiling system; anda quantum processing unit comprising a control system coupled to a plurality of quantum devices,wherein the compiling system is configured to:receive a quantum state specification comprising a plurality of state variables defining a quantum state;determine a compressed representation of the quantum state, wherein determining the compressed representation of the quantum state comprises:splitting the plurality of state variables into a plurality of state variable subsets; anddetermining a respective polynomial approximation representing each respective subset of the plurality of state variable subsets, andsend the compressed representation to the control system, andwherein the control system is configured to:receive the compressed representation of the quantum state; anduse the compressed representation to prepare the plurality of quantum devices according to the quantum state.

2. The quantum computing system of claim 1, wherein the plurality of state variables has an associated size value, and wherein a size of each of the plurality of state variable subsets is equal to the size value.

3. The quantum computing system of claim 1 or claim 2, wherein the quantum devices are qubits.

4. The quantum computing system of any preceding claim, wherein all respective polynomial approximations have equal polynomial degree.

5. The quantum computing system of any preceding claim, wherein each polynomial approximation comprises a respective plurality of polynomial coefficients.

6. The quantum computing system of any preceding claim, wherein a total number of polynomial coefficients associated with all respective polynomial approximations is less than a total number of state variables in the plurality of state variables.

7. The quantum computing system of any preceding claim, wherein all respective state variable subsets have equal size.

8. The quantum computing system of any preceding claim, wherein each respective state variable subset comprises a plurality of sequential state variables.

9. A method of initialising a plurality of quantum devices in a quantum computing system, the method comprising:receiving, at a compiling system of the quantum computing system, a quantum state specification comprising a plurality of state variables defining a quantum state;determining, at the compiling system, a compressed representation of the quantum state, wherein determining the compressed representation of the quantum state comprises:splitting the plurality of state variables into a plurality of state variable subsets; anddetermining a respective polynomial approximation representing each respective subset of the plurality of state variable subsets,sending the compressed representation from the compiling system to a control system of the quantum computing system; andusing the compressed representation, at the control system, to prepare the plurality of quantum devices according to the quantum state.

10. The method of claim 9, wherein the plurality of state variables has an associated size value, and wherein a size of each of the plurality of state variable subsets is equal to the size value.

11. The method of claim 9 or claim 10, wherein the quantum devices are qubits.

12. The method of any of claims 9 to 11, wherein all respective polynomial approximations have equal polynomial degree.

13. The method of any of claims 9 to 12, wherein each polynomial approximation comprises a respective plurality of polynomial coefficients.

14. The method of any of claims 9 to 13, where a total number of polynomial coefficients associated with all respective polynomial approximations is less than a total number of state variables in the plurality of state variables.

15. The method of any of claims 9 to 14, wherein all respective state variable subsets have equal size.

16. The method of any of claims 9 to 15, wherein each respective state variable subset comprises a plurality of sequential state variables.

17. A computer-readable medium comprising instructions which, when executed by a quantum computing system, cause the quantum computing system to carry out the method of any of claims 9 to 16.

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