Optimal calibration of gates in quantum computing systems

Abstract

A method for performing a quantum computing process includes: mapping a plurality of logical qubits to a plurality of physical qubits of a quantum processor using a classical computer, such that a plurality of quantum circuits can be executed using the physical qubits of the quantum processor, and minimizing the total distortion of the plurality of quantum circuits, wherein each physical qubit includes a trapped ion, and each of the plurality of quantum circuits includes a plurality of single-qubit gates and a plurality of two-qubit gates within the plurality of logical qubits; calibrating the two-qubit gates within a first plurality of pairs of physical qubits using a system controller, thereby reducing the distortion of the two-qubit gates within the first plurality of pairs of physical qubits; and applying laser light to the plurality of physical qubits. A laser pulse, each laser pulse causing single-qubit gate operations and two-qubit gate operations in each of the plurality of quantum circuits, executes the plurality of quantum circuits on the quantum processor; after the plurality of quantum circuits are executed on the quantum processor, the population of the qubit states of the physical qubits in the quantum processor is measured by the system controller; and the measured population of the qubit states of the physical qubits is output by the classical computer as the execution result of the plurality of quantum circuits, wherein the execution result of the plurality of quantum circuits is configured to be displayed on a user interface, stored in the memory of the classical computer, or transferred to another computing device.

Description

[0001] Government licensing rights

[0002] This invention was completed with the support of the U.S. government, as granted patent number 70NANB 16H168 by the National Institute of Standards and Technology (NIST). The U.S. government holds certain rights to this invention. Technical Field

[0003] This disclosure generally relates to a method for performing computations in a quantum computing system, and more specifically, to a method for optimizing resources for calibrating quantum gate operations to perform a series of quantum gate operations within a quantum computing system comprising a set of trapped ions. Background Technology

[0004] In the physical systems proposed for building large-scale quantum computers, there is a group of ions (e.g., charged atoms) trapped and suspended in a vacuum by an electromagnetic field. These ions possess internal hyperfine states, separated by frequencies in the range of several GHz, and can be used as computational states for qubits (called "qubit states"). These hyperfine states can be controlled using radiation provided by a laser, or sometimes referred to herein as interaction with a laser beam. Using this laser interaction, ions can be cooled to near their motional ground state. Ions can also be optically pumped with high precision to one of two hyperfine states (qubit preparation), manipulated between two hyperfine states by a laser beam (single-qubit gate operation), and their internal hyperfine states detected by fluorescence when a resonant laser beam is applied (readout qubit). A pair of ions can be controllably entangled using a laser pulse through a force dependent on the qubit state (two-qubit gate operation), which couples the ions to a collective motion mode of a group of trapped ions generated by inter-ion Coulomb interactions. Entanglement typically occurs when pairs or groups of ions (or particles) are generated, interact, or come into spatial proximity such that the quantum state of each ion cannot be described independently of the quantum states of other ions, even when the ions are far apart.

[0005] In such quantum computing systems, quantum computation can be performed by executing a set of single-qubit gate operations and two-qubit gate operations. Although methods for applying these fundamental building blocks of quantum computing have been established, control errors exist in the hardware of quantum computing systems due to miscalibration of control parameters (such as the frequency or amplitude of the laser pulse to be applied to the qubits). These control errors are primarily due to a lack of knowledge about how ions will interact and the characteristics of the quantum computing hardware within the system. Therefore, it is necessary to correct (i.e., calibrate) the control parameters in quantum computing systems to perform reliable and scalable quantum computation. However, calibration typically requires repeated measurements of the qubits to collect statistics across a considerable parameter space of the quantum computing system's control parameters. Therefore, calibration can be an expensive and time-consuming task.

[0006] Therefore, a method is needed to minimize the resources used for calibrating control parameters within an acceptable error range in quantum computing. Summary of the Invention

[0007] Embodiments of this disclosure provide a method for performing a quantum computing process. The method includes: mapping a plurality of logical qubits to a plurality of physical qubits of a quantum processor using a classical computer, such that a plurality of quantum circuits can be executed using the physical qubits of the quantum processor, and minimizing the total distortion of the plurality of quantum circuits, wherein each physical qubit includes a trapped ion, and each of the plurality of quantum circuits includes a plurality of single-qubit gates and a plurality of two-qubit gates within the plurality of logical qubits; calibrating the two-qubit gates in a first plurality of pairs of physical qubits using a system controller, thereby reducing the distortion of the two-qubit gates in the first plurality of pairs of physical qubits; and applying laser pulses to the plurality of physical qubits, each laser pulse... Each of the plurality of quantum circuits induces single-qubit gate operations and two-qubit gate operations to execute the plurality of quantum circuits on the quantum processor; after the plurality of quantum circuits are executed on the quantum processor, the population of the qubit states of the physical qubits in the quantum processor is measured by the system controller; and the measured population of the qubit states of the physical qubits is output by the classical computer as the execution result of the plurality of quantum circuits, wherein the execution result of the plurality of quantum circuits is configured to be displayed on a user interface, stored in the memory of the classical computer, or transferred to another computing device.

[0008] Embodiments of this disclosure also provide a quantum computing system. The quantum computing system includes: a quantum processor comprising a plurality of physical qubits, wherein each physical qubit comprises a trapped ion; a classical computer configured to map a plurality of logical qubits to the plurality of physical qubits such that a plurality of quantum circuits can be executed using the physical qubits, and the total distortion of the plurality of quantum circuits is minimized, wherein each of the plurality of quantum circuits includes a plurality of single-qubit gates and a plurality of two-qubit gates within the plurality of logical qubits; and a system controller configured to calibrate two-qubit gates within a first plurality of pairs of physical qubits such that the distortion of the two-qubit gates within the first plurality of pairs of physical qubits is reduced by means of the plurality of... A laser pulse is applied to a physical qubit, each laser pulse causing a single-qubit gate operation and a two-qubit gate operation in each of the plurality of quantum circuits to execute the plurality of quantum circuits on the quantum processor, and after the plurality of quantum circuits are executed on the quantum processor, the population of the qubit state of the physical qubit in the quantum processor is measured, wherein the classical computer is further configured to output the measured population of the qubit state of the physical qubit as the execution result of the plurality of quantum circuits, wherein the execution result of the plurality of quantum circuits is configured to be displayed on a user interface, stored in the memory of the classical computer, or transferred to another computing device.

[0009] Embodiments of this disclosure further provide a quantum computing system including a non-volatile memory storing a plurality of instructions. When executed by one or more processors, the plurality of instructions cause the quantum computing system to perform operations including: mapping a plurality of logical qubits from a classical computer to a plurality of physical qubits of a quantum processor, such that a plurality of quantum circuits can be executed using the physical qubits of the quantum processor, and minimizing the total distortion of the plurality of quantum circuits, wherein each physical qubit includes a trapped ion, and each of the plurality of quantum circuits includes a plurality of single-qubit gates and a plurality of two-qubit gates within the plurality of logical qubits; calibrating the two-qubit gates within a first plurality of pairs of physical qubits by a system controller, thereby reducing the distortion of the two-qubit gates within the first plurality of pairs of physical qubits; and by... Laser pulses are applied to physical qubits, each laser pulse causing single-qubit and two-qubit gate operations in each of the plurality of quantum circuits to execute the plurality of quantum circuits on the quantum processor; after the plurality of quantum circuits are executed on the quantum processor, the population of the qubit states of the physical qubits in the quantum processor is measured by the system controller; and the measured population of the qubit states of the physical qubits is output by the classical computer as the execution result of the plurality of quantum circuits, wherein the execution result of the plurality of quantum circuits is configured to be displayed on a user interface, stored in the memory of the classical computer, or transmitted to another computing device. Attached Figure Description

[0010] To gain a detailed understanding of the above-described features of the invention, a more specific description of the invention, briefly summarized above, can be obtained by referring to the embodiments, some of which are illustrated in the accompanying drawings. However, it should be noted that the drawings only illustrate typical embodiments of the invention and should not be considered as limiting the scope of the invention, as the invention allows for other equally effective embodiments.

[0011] Figure 1 This is a schematic partial view of an ion trap quantum computing system according to one embodiment.

[0012] Figure 2 A schematic diagram of an ion trap for confining ions in a set according to one embodiment is depicted.

[0013] Figure 3 A schematic energy diagram of each ion in a set of trapped ions according to one embodiment is depicted.

[0014] Figure 4 The quantum state of an ion, represented as a point on the surface of a Bloch sphere, is described.

[0015] Figure 5A, Figure 5B and Figure 5C Some schematic collective lateral motion patterns of five groups of trapped ions were depicted.

[0016] Figure 6A and Figure 6B A schematic diagram depicting the motion sideband spectrum and motion pattern of each ion according to one embodiment is provided.

[0017] Figure 7 An example of pairwise maximum common edge subgraphs (MCEs) according to one embodiment is depicted.

[0018] Figure 8 A flowchart is depicted illustrating a method for computing the most compact cumulative hypergraph (MCCS) using an approximate MCCS algorithm according to one embodiment.

[0019] Figure 9 A flowchart is depicted for a method 900 for computing the most compact cumulative hypergraph (MCCS) using a genetic algorithm according to one embodiment.

[0020] Figure 10A and Figure 10B Example simulation results depicting the reduction in average fidelity and calibration budget requirements of the input quantum circuit according to one embodiment are presented.

[0021] Figure 11A and Figure 11B Example simulation results depicting the reduction in average fidelity and calibration budget requirements of the input quantum circuit according to one embodiment are presented.

[0022] Figure 12A and Figure 12B An example simulation of reduced search function call counts and calibration budget requirements according to one embodiment is presented.

[0023] For ease of understanding, the same reference numerals are used where possible to indicate the same elements common in the figures. In the figures and the following description, an orthogonal coordinate system including the X, Y, and Z axes is used. For convenience, it is assumed that the direction indicated by the arrows in the figures is positive. It is contemplated that elements disclosed in some embodiments may be advantageously used in other embodiments without specific description. Detailed Implementation

[0024] The embodiments described herein generally relate to methods for performing computations in a quantum computing system, and more specifically, to methods for optimizing resources required to perform a series of quantum gate operations in a quantum computing system comprising a set of trapped ions. The method may include a process for calibrating aspects of the quantum gate operations used in the computations performed by the quantum computing system.

[0025] Embodiments of this disclosure include a quantum computing system capable of performing quantum computing processes using a classical computer, a system controller, and a quantum processor. The classical computer performs supporting tasks, including selecting a quantum algorithm to use, computing a quantum circuit to run the quantum algorithm, and outputting the execution results of the quantum circuit using a user interface. Software programs for performing these tasks are stored in non-volatile memory within the classical computer. The quantum processor includes trapped ions coupled to various hardware components, including lasers for manipulating the internal hyperfine states (qubit states) of the trapped ions and photomultiplier tubes (PMTs) for reading out the internal hyperfine states (qubit states) of the trapped ions. The system controller receives instructions from the classical computer for controlling the quantum processor and controls various hardware components associated with and controlling any and all aspects for executing the instructions for controlling the quantum processor and sending readouts from the quantum processor and the output of the results read out therefrom to the classical computer. In some embodiments, the classical computer then uses the computational results based on the output of the readout results to form a result set, which is then provided to the user in the form of results displayed on a user interface, stored in memory, and / or transferred to another computing device to solve a technical problem.

[0026] I. General Hardware Configuration

[0027] Figure 1 This is a partial schematic diagram of an ion trap quantum computing system according to one embodiment. The ion trap quantum computing system 100 includes a classical (digital) computer 102, a system controller 104, and quantum processors, which are trapped ions 106 (i.e., five shown) extending along the Z-axis. Each ion in the trapped ion group 106 is an ion having a nuclear spin I and an electron spin S, the difference between the nuclear spin I and the electron spin S being zero, such as a ytterbium ion. 171 Yb + Barium ions 133 Ba + cadmium ions 111 Cd + or 113 Cd + They all possess nuclear spin. and 2 S 1 / 2 Hyperfine state. In some embodiments, all ions in the trapped ion group 106 are of the same species and isotope (e.g., 171 Yb + In some other embodiments, the trapped ion assembly 106 comprises one or more species or isotopes (e.g., some ions are...). 171 Yb + Some other ions are 133 Ba +In another embodiment, the trapped ion group 106 may include various isotopes of the same species (e.g., different isotopes of Yb, different isotopes of Ba). Ions in the trapped ion group 106 are individually addressed using separate laser beams. The classical computer 102 includes a central processing unit (CPU), memory, and support circuitry (or I / O). The memory is connected to the CPU and may be one or more readily available memories, such as read-only memory (ROM), random access memory (RAM), floppy disk, hard disk, or any other form of local or remote digital storage. Software instructions, algorithms, and data may be encoded and stored in the memory to instruct the CPU. Support circuitry (not shown) is also connected to the CPU to support the processor in a conventional manner. Support circuitry may include conventional caches, power supplies, clock circuits, input / output circuits, subsystems, etc.

[0028] Imaging objectives 108, such as those having a numerical aperture (NA) of, for example, 0.37, collect fluorescence from the ions along the Y-axis and map each ion onto a multichannel photomultiplier tube (PMT) 110 for individual ion measurement. A non-co-propagating Raman laser beam from laser 112, provided along the X-axis, operates on the ions. A diffraction beam splitter 114 creates a static Raman beam array 116, switched separately using a multichannel acousto-optic modulator (AOM) 118, and is configured to selectively act on individual ions. A global Raman laser beam 120 is configured to irradiate all ions at once. In some embodiments, individual Raman laser beams (not shown) irradiate individual ions separately. A system controller (also referred to as an “RF controller”) 104 controls the AOM 118 and thus controls the laser pulses to be applied to the trapped ions in the trapped ion group 106. System controller 104 includes a central processing unit (CPU) 122, read-only memory (ROM) 124, random access memory (RAM) 126, and storage unit 128, etc. CPU 122 is the processor of system controller 104. ROM 124 stores various programs, and RAM 126 is the working memory for various programs and data. Storage unit 128 includes non-volatile memory, such as hard disk drive (HDD) or flash memory, and stores various programs even when power is off. CPU 122, ROM 124, RAM 126, and storage unit 128 are interconnected via bus 130. System controller 104 executes control programs stored in ROM 124 or storage unit 128 and uses RAM 126 as its working area. Control programs include software application programs that include program code executable by the processor to perform various functions associated with receiving and analyzing data and controlling any and all aspects of the methods and hardware used to create the ion trap quantum computer system 100 discussed herein.

[0029] Figure 2A schematic diagram of an ion trap 200 (also referred to as a "Paul trap") for confining ions in group 106 according to one embodiment is depicted. The confinement potential is applied by both a static (DC) voltage and a radio frequency (RF) voltage. The static (DC) voltage V... S Applications are made to end cap electrodes 210 and 212 to confine ions along the Z-axis (also referred to as the "axial" or "longitudinal" axis). Due to the Coulomb interactions between the ions, the ions in group 106 are nearly uniformly distributed axially. In some embodiments, ion trap 200 includes four hyperbolic electrodes 202, 204, 206, and 208 extending along the Z-axis.

[0030] During operation, a sinusoidal voltage V1 (with amplitude V) is applied. RF / 2) Apply a sinusoidal voltage V2 (with amplitude V1) to a pair of opposing electrodes 202, 204, and phase-shift the sinusoidal voltage V1 by 180°. RF / 2) Driven frequency ω RF A quadrupole potential is generated by applying a sinusoidal voltage to the opposite pair of electrodes 206, 208. In some embodiments, a sinusoidal voltage is applied only to the opposite pair of electrodes 202, 204, and the opposite pair of electrodes 206, 208 is grounded. The quadrupole potential generates an effective confinement force for each trapped ion in the XY plane perpendicular to the Z-axis (also referred to as "radial" or "lateral"), which is proportional to the distance from the saddle point (i.e., the position in the axial (Z direction) direction) where the RF electric field disappears. The radial (i.e., the direction in the XY plane) motion of each ion is approximately resonant (referred to as long-term motion), and the restoring force is radially directed toward the saddle point and can be transmitted through the spring constant k. x and k y The modeling is described in more detail below. In some embodiments, when the quadrupole potential is radially symmetrical, the radial spring constant is modeled as equal. However, in some undesirable cases, the radial motion of ions may be distorted due to some asymmetry in the physical trap configuration, small DC patch potentials due to inhomogeneities on the electrode surfaces, etc., and due to these and other external sources of distortion, the center of the ions may deviate from the saddle point.

[0031] Figure 3 A schematic energy diagram 300 depicts each ion in a trapped ion array 106 according to one embodiment. Each ion in the trapped ion array 106 is an ion having a nuclear spin I and an electron spin S, the difference between the nuclear spin I and the electron spin S being zero. In one example, each ion may be a positive ytterbium ion. 171 Yb + It has nuclear spin and 2 S 1 / 2 Hyperfine states (i.e., two electronic states) have energy splitting corresponding to ω. 01 / 2π = 12.642821 GHz of frequency difference (referred to as the "carrier frequency"). In other examples, each ion could be a positive barium ion. 133 Ba + cadmium ions 111 Cd + or 113 Cd + They all possess nuclear spin. and 2 S 1 / 2 Hyperfine states. A qubit consists of two hyperfine states, denoted as |0> and |1>, where the hyperfine ground state (i.e., ...) is chosen. 2 S 1 / 2 |0> is represented by a low-energy state in a hyperfine state. In the following text, the terms "hyperfine state," "internal hyperfine state," and "qubit" are used interchangeably to denote |0> and |1>. Each ion can be cooled (i.e., its kinetic energy can be reduced) to a phonon-free ground state |0> close to any motion mode m by known laser cooling methods, such as Doppler cooling or resolved sideband cooling. m (that is, n) ph =0), and then optical pumping is used to prepare qubit states in the hyperfine ground state |0>. Here, |0> represents the individual qubit state of the trapped ion, while |0> with subscript m m The ground state of motion represents the motion mode m of the trapped ion group 106.

[0032] The individual qubit state of each trapped ion can be generated by, for example, a 355 nm mode-locked laser via excitation. 2 P 1 / 2 Manipulated by energy levels (denoted as |e>). Figure 3 As shown, the laser beam from the laser can be split into a pair of non-co-propagating laser beams (a first laser beam with frequency ω1 and a second laser beam with frequency ω2) in a Raman configuration, and with respect to the transition frequency ω between |0> and |e>. 0e With the single-photon transition detuning frequency Δ=ω1-ω 0e Disharmony, such as Figure 3 As shown. The two-photon transition detuning frequency δ includes the amount of energy supplied to the trapped ion by the first and second laser beams, which, upon recombination, is used to transfer the trapped ion between the hyperfine states |0> and |1>. When the single-photon transition detuning frequency Δ is much greater than the two-photon transition detuning frequency (also simply referred to as the "detuning frequency") δ = ω1 - ω2 - ω 01 (Hereinafter expressed as ±μ, where μ is a positive value), single-photon Rabi frequency Ω 0e (t) and Ω 1e(t) (which is time-dependent and determined by the amplitude and phase of the first and second laser beams, with Rabi oscillations occurring at this frequency between states |0> and |e> and between states |1> and |e>, respectively), and the spontaneous emission rate from the excited state |e>, induces Rabi oscillations (called "carrier transitions") between the two hyperfine states |0> and |1> at the two-photon Rabi frequency Ω(t). The intensity (i.e., the absolute value of the amplitude) of the two-photon Rabi frequency Ω(t) is related to Ω. 0e Ω 1e / 2Δ is proportional, where Ω 0e and Ω 1e These are the single-photon Rabi frequencies generated by the first and second laser beams, respectively. In the following text, this set of non-co-propagating laser beams in the Raman configuration used to manipulate the internal hyperfine state (qubit state) of a qubit may be referred to as a "composite pulse" or simply a "pulse," and the time-dependent pattern of the resulting two-photon Rabi frequency Ω(t) may be referred to as the "amplitude" of the pulse or simply a "pulse," which will be explained and further described below. Detuning frequency δ=ω1-ω2-ω 01 The detuning frequency of the composite pulse or the detuning frequency of the pulse can be referred to as the "amplitude" of the composite pulse. The amplitude of the two-photon Rabi frequency Ω(t), determined by the amplitudes of the first and second laser beams, can be referred to as the "amplitude" of the composite pulse.

[0033] It should be noted that the specific atomic species used in the discussion presented herein are merely one example of atomic species that have a stable and well-defined two-level energy structure upon ionization and possess optically accessible excited states, and are therefore not intended to limit the possible configurations, specifications, etc., of the ion trap quantum computer according to the present invention. Other ion species include, for example, alkaline earth metal ions (Be...). + Ca + 、Sr + Mg + Ba + ) or transition metal ions (Zn + Hg + Cd + ).

[0034] supply Figure 4To aid in visualizing the qubit states of ions, they are represented as points on the surface of a Bloch sphere 400 with an azimuth angle φ and a polar angle θ. The application of the composite pulse described above results in Rabi oscillations between qubit states |0> (represented as the north pole of the Bloch sphere) and |1> (the south pole of the Bloch sphere). Adjusting the duration and amplitude of the composite pulse flips the qubit state from |0> to |1> (i.e., from the north pole to the south pole of the Bloch sphere), or from |1> to |0> (i.e., from the south pole to the north pole of the Bloch sphere). This application of the composite pulse is called a “π-pulse”. Furthermore, by adjusting the duration and amplitude of the composite pulse, the qubit state |0> can be converted into a superposition state |0>+|1>, where the two qubit states |0> and |1> are added with equal weight and in the same phase (for convenience, the normalization factor of the superposition state is omitted below), and the qubit state |1> can be converted into a superposition state |0>-|1>, where the two qubit states |0> and |1> are added with equal weight and out of phase. This application of the composite pulse is called the “π / 2-pulse”. More generally, the superposition of the two equally weighted qubit states |0> and |1> is represented by a point located on the equator of the Bloch sphere. For example, the superposition state |0>±|1> corresponds to points with an azimuth angle φ of zero and π above the equator, respectively. The superposition state corresponding to a point with an azimuth angle φ on the equator is represented as |0>+e iφ |1> (for example, for φ=±π / 2, it is |0>±i|1>). The transformation between two points on the equator (i.e., rotation about the Z-axis on the Bloch sphere) can be achieved by shifting the phase of the composite pulse.

[0035] II. Entanglement Formation

[0036] Figure 5A , Figure 5B and Figure 5C Some schematic structures depict, for example, the collective lateral motion pattern (also referred to simply as the "motion pattern structure") of a group 106 of five trapped ions. Here, due to the static voltage V applied to the end cap electrodes 210 and 212... S The resulting confinement potential is weaker compared to the radial confinement potential. The collective motion mode of the trapped ion group 106 in the transverse direction is determined by the combination of the Coulomb interaction between the trapped ions and the confinement potential generated by the ion trap 200. The trapped ions undergo collective transverse motion (referred to as "collective transverse motion mode", "collective motion mode", or simply "motion mode"), where each mode has a different energy (or equivalent frequency) associated with it. The motion mode with the m-th lowest energy is referred to below as |n ph > m , where n phThe number of motion quanta (called phonons in units of energy excitation) in a motion mode is represented by M, and the number of motion modes M in a given transverse direction is equal to the number of ions trapped in group 106. Figures 5A-5C The illustration shows examples of different types of collective lateral motion patterns that the five trapped ions in group 106 may experience. Figure 5A It is a common motion mode with the highest energy |n ph > M Where M is the number of motion patterns. In common motion patterns |n> M In this system, all ions oscillate in phase laterally. Figure 5B It is the tilting motion mode with the second highest energy |n ph > M-1 In the tilted motion mode, the ions at both ends move out of phase (i.e., in opposite directions) in the lateral direction. Figure 5C It is a higher-order motion mode |n ph > M-3 The diagram shows that its energy is lower than that of the tilting motion mode |n ph > M-1 The energy in which ions move in more complex patterns.

[0037] It should be noted that the specific configuration described above is only one of several possible examples of traps for confining ions according to this disclosure, and does not limit the possible configurations, specifications, etc., of traps according to this disclosure. For example, the geometry of the electrodes is not limited to the hyperbolic electrodes described above. In other examples, the trap that generates an effective electric field so that the movement of ions in the radial direction is resonant can be a multilayer trap, in which multiple electrode layers are stacked and an RF voltage is applied to two diagonally opposite electrodes, or a surface trap, in which all electrodes are located in a single plane on the chip. Furthermore, the trap can be divided into multiple segments, and adjacent pairs of segments can be connected by shuttles of one or more ions or coupled by photonic interconnects. The trap can also be an array of single confinement regions closely arranged on a microfabricated ion trap chip. In some embodiments, in addition to the RF component described above, the quadrupole potential also has a spatially varying DC component.

[0038] In ion trap quantum computers, motion patterns can serve as a data bus to mediate entanglement between two qubits, which is then used to perform XX-gate operations. That is, each of the two qubits is entangled with a motion pattern, and this entanglement is then transferred to entanglement between the two qubits using motion sideband excitations, as described below. Figure 6A and Figure 6B The schematic depiction illustrates a scenario according to one embodiment where a frequency ω is... m Movement patterns | n ph > MA view of the motion sideband spectrum of ions in group 106. (See image.) Figure 6B As shown, when the detuning frequency of the composite pulse is zero (i.e., the frequency difference between the first and second laser beams is tuned to the carrier frequency, δ=ω1-ω2-ω), 01 When |ω1| = 0, a simple Rabi oscillation (carrier transition) occurs between the qubit states |0> and |1>. When the detuning frequency of the composite pulse is positive (i.e., the frequency difference between the first and second laser beams is tuned to be higher than the carrier frequency, δ = ω1 - ω2 - ω1), a simple Rabi oscillation (carrier transition) occurs. 01 =μ>0, called the blue sideband), in the composite qubit motion state |0>|n ph > m and |1>|n ph +1> m Rabi oscillations occur between (i.e., when the qubit state |0> flips to |1>, an oscillation occurs from |n>) ph > m The m-th motion mode excited by n phonons is represented as |n ph +1> m Having (n ph +1) Transition of the m-th motion mode excited by phonons). When the detuning frequency of the composite pulse is negative (i.e., the frequency difference between the first and second laser beams is tuned to be lower than the carrier frequency of motion mode |n) ph > m frequency ω m δ=ω1-ω2-ω 01 =-μ<0, called the red sideband), in the composite qubit motion state |0>|n ph > m and |1>|n ph -1> m Rabi oscillations occur between (i.e., when the qubit state |0> flips to |1>, a transition from motion mode |n> occurs). ph > m Motion patterns excited by one less phonon | n ph -1> m (The transition). A π / 2 pulse applied to the blue sideband of the qubit will recombine the qubit motion state |0>|n. ph > m Convert to |0>|n ph > m and |1>|n ph +1> m The superposition of the qubits. A π / 2 pulse applied to the red sideband of the qubit will cause the composite qubit to move |0>|n. ph > m Convert to |0>|n ph > m and |1>|n ph-1> m The superposition of two-photon Rabi frequencies. When the two-photon Rabi frequency Ω(t) is greater than the detuning frequency δ=ω1-ω2-ω 01 =±μ hours, blue sideband transitions or red sideband transitions can be selectively driven. Therefore, by applying an appropriate type of pulse, such as a π / 2 pulse, a qubit can be entangled with a desired motion pattern, which can then be entangled with another qubit, resulting in entanglement between the two qubits, which is required to perform XX-gate operations in an ion trap quantum computer.

[0039] By controlling and / or guiding the transitions of the combined qubit states as described above, XX-gate operations can be performed on two qubits (the i-th and j-th qubits). Generally, the XX-gate operation (with maximum entanglement) transitions the two qubit states as follows: |0> i |0> j ,|0> i |1> j ,|1> i |0> j and |1> i |1> j :

[0040]

[0041] For example, when two qubits (the i-th and j-th qubits) are initially in the hyperfine ground state |0> (denoted as |0> i |0> j ), and subsequently when a π / 2 pulse on the blue sideband is applied to the i-th qubit, the composite state of the i-th qubit and the motion mode is |0> i |n ph > m Converted to |0> i |n ph > m and |1> i |n ph +1> m The superposition of the two qubits and the motion mode transforms the composite state into |0> i |0> j |n ph > m and |1> i |0> j |n ph +1> m The superposition of the two states. When a π / 2 pulse on the red sideband is applied to the j-th qubit, the composite state of the j-th qubit and the motion mode is |0> j |n ph > m Convert to |0> j |nph > m and |1> j |n ph -1> m The superposition of, and the combinational state |0> j |n ph +1> m Convert to |0> j |n ph +1> m and |1> j |n ph > m The superposition of.

[0042] Therefore, applying a π / 2 pulse on the blue sideband at the i-th qubit and a π / 2 pulse on the red sideband at the j-th qubit can transform the composite state |0> of the two qubits and the motion mode. i |0> j |n ph > m Convert to |0> i |0> j |n ph > m and |1> i |1> j |n ph > m The two qubits are now in an entangled state due to the superposition of their components. It should be clear to those skilled in the art that the number of phonon excitations relative to the initial number of phonon excitations, n, can be removed using a sufficiently complex pulse sequence. ph Two qubit states entangled in different motion modes (i.e., |1> i |0> j |n ph +1> m and |0> i |1> j |n ph -1> m Therefore, the composite state of the two qubits and the motion mode after the XX-gate operation can be considered unentangled, because at the end of the XX-gate operation, the number of initial phonon excitations n in the m-th motion mode is reduced. ph The state remains unchanged. Therefore, the following description generally describes the state of the qubit before and after the XX gate operation, without including the motion mode.

[0043] More generally, by applying an amplitude Ω (i) The complex state of the i-th and j-th qubits, transformed by pulses on the sidebands with a detuning frequency μ and a duration τ (called the "gate duration"), can be determined based on the entanglement interaction χ. (i，j) (τ) is described as follows:

[0044] |0> i |0> j →cos(2χ (i，j) (τ))|0) i |0> j -i sin(2χ (i，j) (τ))|1> i |1> j

[0045] |0> i |1> j →cos(2χ (i，j) (τ))|0> i |1> j -i sin(2χ (i，j) (τ))|1> i |0> j

[0046] |1> i |0> j →-i sin(2χ (i，j) (τ))|0> i |1> nj +cos(2χ (i，j) (τ))|1> i |0> j

[0047] |1> i |1> j →-i sin(2χ (i，j) (τ))|0> i |0> j +cos(2χ (i，j) (τ))|1> i |1> j

[0048] in,

[0049]

[0050] It is to quantify the i-th ion and have a frequency ω m The Lamb-Dicke parameter is the coupling strength between the m-th motion modes, and M is the number of motion modes (equal to the number of ions N in group 106).

[0051] The entangled interaction between the two qubits described above can be used to perform XX-gate operations. XX-gate operations (XX gates) together with single-qubit gate operations (R gates) form a gate set {R, XX}, which can be used to construct a quantum computer configured to perform the desired computational process. Among the several known sets of logic gates that can decompose any quantum algorithm, a set of logic gates, typically denoted as {R, XX}, is inherent to the quantum computing system of trapped ions described in this paper. Here, R gates correspond to the manipulation of the single-qubit state of a trapped ion, while XX gates (also called “entanglement gates”) correspond to the manipulation of the entanglement of two trapped ions.

[0052] To perform the XX gate operation between the i-th and j-th qubits, construct a structure that satisfies condition χ. (i，j) (τ)=θ (i，j) (0<θ (i，j) ≤π / 8)(i.e., entanglement correlation χ (i，j) (τ) has an expected value θ (i，j) A pulse (called the condition for non-zero entanglement interaction) is applied to the i-th and j-th qubits. When θ (i，j) When π / 8, the transition of the composite state of the i-th and j-th qubits corresponds to the XX-gate operation with maximum entanglement. The amplitude Ω of the pulse to be applied to the i-th and j-th qubits... (i) (t) and Ω (j) (t) is a control parameter that can be adjusted to ensure non-zero tunable entanglement between the i-th and j-th qubits, thereby performing the desired XX gate operation on the i-th and j-th qubits.

[0053] III. Calibration

[0054] Quantum computing can be performed in quantum computing systems such as the ion trap quantum computing system 100 using a set of quantum gate operations, including single-qubit gate operations (R-gates) and two-qubit gate operations, such as XX-gate operations (XX-gates). Although methods for applying these fundamental building blocks of quantum computing have been established, control errors exist, caused by miscalibration of control parameters in the hardware of the quantum computing system. These control errors are primarily due to a lack of knowledge about how ions and qubits operate during quantum gate operations in the quantum computing system. Therefore, calibration—the task of understanding and adjusting the control parameters in the quantum computing system to correct control errors—is required to provide scalable and reliable quantum computing results.

[0055] Calibration processes typically require repeated measurements of qubits to collect statistics across a considerable parameter space of the control parameters of a quantum computing system. Therefore, calibration can be an expensive and time-consuming task. For example, if all gate operations are calibrated, the sequence of calibration steps or procedures increases quadratically with the number of qubits in the quantum computing system, potentially degrading the quality of the calibration process. Therefore, it is necessary to optimize the sequence of calibration steps or procedures within the acceptable error range of quantum computing.

[0056] In the embodiments described herein, methods are provided for optimizing resources for calibrating quantum gate operations to execute a batch of quantum circuits (i.e., a series of quantum gate operations) on a quantum computing system. In a quantum computing system such as the ion trap quantum computing system 100, single-qubit gate operations (R-gates) can be executed with high precision with minimal calibration effort, thus requiring only significant effort to calibrate two-qubit gate operations, such as XX-gate operations (XX-gates).

[0057] III.A. Calibration Resource Description

[0058] Quantum processors, such as the trapped ion array 106 in the ion trap quantum computing system 100, are derived from the fully connected system diagram G. s =(V s E s , ∈) specifies, where vertex i∈V s Let i represent the physical qubit (i.e., the trapped ion) of the quantum processor. Connect the two vertices i, j ∈ V. s Each edge (i, j) ∈ E s Represents a two-qubit gate between physical qubits i and j. Each edge (i, j) ∈ E s The associated distortion ∈(i,j) of a two-qubit gate with physical qubits i and j. The associated distortion ∈(i,j) is symmetric with respect to the interchange of physical qubits i and j (i.e., ∈(i,j) = ∈(j,i)).

[0059] The calibration of the two-qubit gate in the ion trap quantum computing system 100 includes selecting all edges (i, j) ∈ E s The ion trap quantum computer system 100 calibrates edges (i, j) within a subset S by tuning control parameters associated with the hardware in the subset S. Generally, the associated distortion ∈ (i, j) of an edge (i, j) within a calibrated subset S is lower than the associated distortion ∈ (i, j) of an edge (i, j) outside a subset S that has not yet been calibrated. Calibration is performed on the subset S (hence also referred to as the "calibration set"), and therefore, calibration is optimized if the subset S has a minimum size within an acceptable error range.

[0060] Quantum computing in the ion trap quantum computing system 100 is achieved by executing a batch of... It is executed by quantum circuits. From circuit diagram G c =(V c E c w) specifies that vertex k∈V c Let k represent a logical qubit. Each logical qubit k and l are connected by two vertices k, l ∈ V. c The edge e = (k, l) ∈ E c This represents a two-qubit gate between two logical qubits k and l. Assume the number of logical qubits in a quantum processor 106 is |V0|0. c |Number of physical qubits or less|V s |. Each edge e = (k, l) ∈ E of logical qubits k and l. c Each has an associated weight w(k, l), which represents the quantum circuit. The number of occurrences of the two-qubit gate between logical qubits k and l. Each logical qubit k is mapped to a physical qubit i, and the mapping between logical and physical qubits is described below. Each quantum circuit A collection of single-qubit and two-qubit gates mapped onto physical qubit i, which can be executed on a quantum processor of a trapped ion array, such as in an ion trap quantum computing system 100, by applying appropriate laser pulses to physical qubit i by a system controller (e.g., system controller 104).

[0061] Given the circuit diagram G c =(V c E c The quantum circuit specified by w) and by system diagram G s =(V s E s The quantum processor specified by ∈ is such that logical qubit k is mapped to physical qubit i, making the quantum circuit... Distortion ε(G) c G s The mapping between physical qubit i and logical qubit k is minimized. In some embodiments, this mapping between physical qubit i and logical qubit k is achieved through a bijective mapping π: V c →V s (i.e., π(G) c ) = G s Through this mapping π, the circuit diagram G c In the context of edge e = (k, l) ∈ E c Mapping to system graph G s The edge π(e) = (π(k), π(l)) ∈ E s(i.e., π(E) c ) = E s The mapping π is calculated such that in the system diagram G... c Quantum circuits executed on Distortion ε(G) c G s Minimize:

[0062]

[0063] Average distortion (i.e., in quantum circuits) The minimum distortion ε(G) on average c G s The following formula is given.

[0064]

[0065] in This batch The number of quantum circuits c. In the embodiments described herein, a specific number Γ (called the calibration budget) of two-qubit gates between physical qubits i and j are selected for calibration, where |S| ≤ Γ. That is, a subset S comprising |S| two-qubit gates is selected for calibration. The subset S is selected such that the average distortion shown in equation (2) is Minimize to

[0066]

[0067] Where ∈ S It is the distortion associated with the two-qubit gates in the calibrated subset S.

[0068] The precise calculation provides the minimum average distortion shown in formula (3). A subset S is computationally difficult, at least NP-hard; therefore, the embodiments described herein provide two heuristic methods for approximating the computation of the subset S. Various notations and definitions used to describe these methods are summarized below.

[0069] First, for simplicity, assume that edge e∈E S All two-qubit gates have equal associated distortion ∈ before calibration. - And after calibration, it has a reduced associated distortion ε + , where ∈ + <∈ - This assumption is referred to below as the binary gate fidelity model.

[0070] Then equation (1) can be simplified to

[0071]

[0072] Therefore, calculating the mapping π is reduced to maximizing the first sum in equation (4). For the predetermined calibration budget Γ, the first sum in equation (4) can be maximized as

[0073]

[0074] Next, the maximum common edge subgraph (MCEs) is introduced because in the circuit diagram G... c In a typical scenario where there may be more edges e than the calibration budget Γ, the calibration set S to be computed is along with a batch The quantum circuit c must have maximum overlap. The MCEs between two unweighted graphs (i.e., the weight w of edge e is 0 or 1) consist of the maximum set of common edges as defined below.

[0075] Definition A.1 (Maximum Common Edge Subgraph) A maximum common edge subgraph identifies isomorphic subgraphs ∩(G1, G2, ..., G...) that have the maximum number of edges in the set of graphs. n ).

[0076] For graph {G1, G2, ... G... N The sequence} is used to determine the maximum common edge subgraph (MCEs) by sequentially computing the paired MCEs, i.e., (((G1∩G2)∩G3)…∩G N This differs from the MCEs of a sequence of graphs defined in Definition A.1. A brute-force approach to finding the MCEs of n graphs requires maximizing the over-permutations of the vertex mappings of all graphs. This scales exponentially with the number of graphs n. Therefore, in the embodiments described herein, the MCEs of n graphs are computed by combining pairs of MCE operations, thereby reducing the complexity to linear with the number of graphs n. However, note that, as Figure 7 As shown, the processes for determining paired MCEs are not correlated. In Figure 7 In the example shown, the pairwise determination of MCEs for the three graphs F, G, and H yields different final results depending on the order in which the pairs of MCEs are composed. Since considering all N! sortings, where N is the total number of graphs for which MCEs are computed, is very expensive, the method described here provides a heuristic for determining the appropriate order of the pairs of MCEs.

[0077] For a weighted graph where each edge has a different weight, such as a circuit graph G c =(V c E c , w), where edge e∈E cGiven an associated weight w(e), the heaviest cumulative subgraph (HCs) is considered to be able to accommodate the weights of the edges. The heaviest cumulative subgraph (HCs) is defined as follows.

[0078] Definition A.2 (Heaviest Cumulative Subgraph) A heaviest cumulative subgraph is a common subgraph of multiple graphs, ∧(G1, G2, ..., G...). n The maximum total weight of the edges is:

[0079]

[0080] in

[0081] The definition of the heaviest cumulative subgraphs (HCs) is closely related to the calibration set S. For example, between two graphs G (a complete graph with two vertices) and H (a complete graph with three vertices and edges, each edge having a different weight), the optimal calibration set S with a calibration budget F = 1 is the HCs between graphs G and H. That is, determining the mapping from G to H maximizes the cumulative edge weights.

[0082] However, in a non-trivial case, the definition of the heaviest cumulative subgraph (HCs) is insufficient to compute the heaviest cumulative subgraph. Therefore, the most compact cumulative supergraph (MCCS) is considered here. This is to minimize the average distortion shown in equation (2). The optimal mapping π can be calculated to make the mapping circuit diagram G c The hypergraph maximizes the sum of the weights of the heaviest Γ edges. By constructing this hypergraph, it will contain a subgraph with Γ edges, which allows mapping to all circuit graphs G. c There is significant overlap. The Most Compact Cumulative Hypergraph (MCCS) is defined as follows. In the binary gate fidelity model, where edge e∈E S All two-qubit gates have equal associated distortion ∈ before calibration. - After calibration, the associated distortion is reduced. + Given circuit diagram G c The MCCS is the solution of the given calibration budget Γ of formula (5).

[0083] Definition A.3 (Most Compact Cumulative Hypergraph) The most compact cumulative hypergraph with respect to the calibration budget Γ is a hypergraph of the graph set, V Γ (G1, G2, ..., G) n The heaviest edge in Γ has the largest sum of weights. This can be obtained by maximizing the total weight of the heaviest edge in Γ as follows:

[0084]

[0085] The sum with superscript is for the first Γ largest elements.

[0086] In some embodiments, circuit diagram G c =(V c E c w) has fewer edges e∈E than the calibration budget Γ c In this case, it is possible to consider including all circuit diagrams G. c A hypergraph with a minimum size. The number of edges in this hypergraph is less than or equal to the calibration budget Γ, and all edges in this hypergraph can be calibrated without considering edge weights. This hypergraph is called the minimum common edge supergraph (mCES) and is defined as follows.

[0087] Definition A.4 (Least Common Hypergraph) The least common hypergraph identification includes elements from the set ∪(G1, G2, ..., G...). n The smallest graph of all graphs.

[0088] III.B. Minimize Calibration Resources

[0089] As mentioned above, for a given calibration budget Γ, the circuit diagram G can be obtained. c The most compact cumulative hypergraph (MCCS) of the set is used to minimize calibration resources. In the embodiments described herein, two heuristic methods for obtaining the MCCS are provided: one is method 800, which calculates the MCCS using an approximate MCCS algorithm, and the other is method 900, which is known in the art, and calculates the MCCS using a genetic algorithm.

[0090] B.1 Approximate MCCS Algorithm

[0091] B.1.1 Algorithm Overview

[0092] Figure 8 Describes the application of an approximate MCCS algorithm to a specified batch quantum circuits Circuit diagram G c =(V c E c The flowchart of method 800 for calculating the most compact cumulative hypergraph (MCCS, hereinafter referred to as M) of the set of w). This hypergraph is used to calculate the circuit diagram G. c =(V c E c The set and system diagram G of w) s The mapping π between them makes the distortion ε(G) shown in equation (2) more accurate. c Gs ) minimized.

[0093] In box 802, a batch of data to be calibrated is received via classical computer 102. quantum circuits

[0094] In box 804, based on a batch received by classical computer 102... quantum circuits To calculate circuit diagram G c =(V c E c The set of (w) in forming the circuit diagram G. c When considering a set of graphs, we must consider graph isomorphism. That is, the set contains circuit graphs G that are not isomorphic to each other. c In the case of weighted graphs, a modified definition of graph isomorphism is used, where two weighted graphs G = (V1, E1, w1) and H = (V2, E2, w2) are isomorphic if there exists a bijective mapping π between the vertex sets of these two graphs, π: V1 → V2, such that the mapping preserves weighted adjacency. edge e∈E c The associated weights w(e) are stored in dictionaries in the non-volatile memory of a classical computer 102.

[0095] In box 806, edge e∈E is computed using classical computer 102. c The number. If all quantum circuits Each edge e∈E c If the quantity is less than or equal to the predetermined calibration budget Γ, then method 800 proceeds to box 808. If edge e∈E c If at least one of the quantities is greater than the predetermined calibration budget Γ, then method 800 proceeds to block 814.

[0096] In box 808, the circuit diagram G is calculated using the classical computer 102. c =(V c E c The minimum common edge hypergraph (mCES) of the set of (w).

[0097] In block 810, the circuit diagram G is calculated using the classical computer 102. c =(V c E cThe method calculates the number of edges in the least common edge hypergraph (mCES) of the matrix (w). If the number of edges in the mCES is less than or equal to a predetermined calibration budget Γ, all edges in the mCES can be calibrated regardless of their associated weights. Then, method 800 proceeds to box 812. If the number of edges in the mCES is greater than the predetermined calibration budget Γ, method 800 proceeds to iteration to compute the MCCS M starting from box 814.

[0098] In box 812, the computed minimum common edge hypergraph (mCES) is output as the computation result by classical computer 102.

[0099] In block 814, the initial iteration of the approximate and iterative calculations of MCCS, M is performed using classical computer 102. In block 814, graph M is initialized as a zero graph (i.e., excluding circuit diagram G). c ), and create circuit diagram G c An ordered list L. The ordered list L is created by sorting the circuit diagrams in descending order of the number of their edges.

[0100] In box 816, the circuit diagram G is calculated using the classical computer 102. c The set is mapped to the system graph G s The mapping π. First, calculate the first circuit diagram G in the ordered list L. c The mapping π (abbreviated as G) is calculated such that circuit diagram G and diagram M overlap. The calculation of mapping π is described in more detail below.

[0101] In box 818, graph G is mapped by mapping π using classical computer 102, and the mapped graph π(G) is merged with graph M, described as follows:

[0102] In box 820, method 800 targets the subsequent circuit diagram G in the ordered list L. c Return to box 816. If all circuit diagrams G in the ordered list L have been considered... c Then, method 800 proceeds to box 822. The resulting diagram M is all the circuit diagrams G in the ordered list L. c The cumulative hypergraph. Figure M is the cumulative hypergraph of all circuit diagrams G. c The approximate MCCS.

[0103] In box 822, among the edges in the cumulative hypergraph M, select the Γ heaviest edges and remove the remaining edges. This final graph is denoted as M. Γ , is an approximate solution that includes the calibration set S as its edges. Once the solution M is identified... Γ For each input circuit, perform one round to M. ΓThe optimal mapping to maximize M Γ And the overlap between each input circuit. This step is performed because the mapping previously used in the construction diagram M is for having M Γ The maximum overlap problem is suboptimal. The method described in this paper uses the mapping identified in this step as the basis for each circuit diagram G. c The final mapping.

[0104] In box 824, the calculated graph M is processed by classical computer 102. Γ The output is the result of the calculation.

[0105] It should be noted that when using the above-described sequential merging method, the circuit diagram G should be arranged as described above. c This may lead to an exact solution for the MCCS.

[0106] B.1.2 Optimal Mapping Algorithm

[0107] In box 806, the mapping π is computed such that graphs G and M overlap. This can be considered an extension of the maximum common edge subgraph (MCEs) problem to weighted graphs. The algorithm for finding the optimal mapping is called a backtracking algorithm, which is similar to the depth-first search (DFS) algorithm known in the art, except that the backtracking algorithm does not involve all possible branches, but backtracks earlier based on a predictor function that checks whether the initial mapping is worthwhile based on the optimization objective. As the basis of the backtracking algorithm, the SplitP algorithm known in the art is used with a modified predictor function. Instead of maximizing the number of common edges in the graph, the cumulative weights of the common subgraphs are maximized to achieve the heaviest cumulative subgraphs (HCs). When multiple potential mappings exist, an additional criterion is used. In this case, the mapping is selected such that the cost of subgraph transformations generated between π(G) and M is minimized. This criterion helps to minimize the number of heaviest edges in the final cumulative supergraph M.

[0108] Definition B.1 (Subgraph Transition Cost): The subgraph transition cost S(G, H) between two weighted graphs G and H is the transition cost of the graph formed by the set of their common edges (see below).

[0109]

[0110] Definition B.2 (Transition Cost): The transition cost T(G, H) between two weighted graphs G and H includes the sum of the weights of added / deleted edges and the weight changes of common edges.

[0111]

[0112] If Then w G(e) = 0. Typically, the cost function f C (w G (e), w H (e) can be w G (e) and w H (e) is a random distance function. In some embodiments, minimum transformation cost is used because, when applied to the unweighted case, it simplifies the mapping problem to a maximum common edge subgraph (MCEs) problem (min π T(π(G), H)=|G|+|H|-2|G∩H|), where |G| represents the number of edges in G.

[0113] Specifically, the mapping algorithm that identifies the optimal mapping of vertices from G to M repeatedly calls the search function. The search function searches for the heaviest cumulative subgraphs (HCs) between G and M with a pre-specified number of edges. If found, the function also returns a vertex mapping from G to M to induce HCs. The maximum possible HCs between G and M can be determined by iteratively reducing the pre-specified number of edges, starting from the smaller of the number of edges in the two input graphs G and M.

[0114] The implementation of the search function involves technical aspects. In short, given two input graphs G and M and a pre-specified number of edges, the function uses a depth-first search to build a mapping, starting with an empty mapping and pairing heuristically selected edges from G with appropriate edges from M at each level of the search tree. The function backtracks if the search in a branch has even a slight chance of exceeding the best result found so far. The function also backtracks if the calculated boundary is less than the pre-specified number of edges. Here, the boundary is calculated as the number of already mapped edges plus the maximum number of edges that could be mapped based on their adjacency with respect to the mapped edges.

[0115] B.1.3 Beam Search

[0116] While the backtracking algorithm used in this paper can be considered a more advanced version of the depth-first search (DFS) algorithm, it still doesn't scale well with the size of the graph. To make the method more scalable, a threshold can be set for the predicted improvement when exploring branches, comparing it to the best improvement found to date. In particular, a stronger condition requires the cumulative weight of the proposed potential subgraph to exceed the best discovery, plus the added threshold. Otherwise, the potential graph will be excluded from further consideration. Multiple branches can be explored further while limiting the number of potential graphs or walkers on the search tree to n, i.e., only the n most promising graphs are retained.

[0117] For a large number of qubits, the circuit graph in this batch may have small heaviest cumulative subgraphs (HCs), which will lead to a rapid burst in the size of the merged graph M. This results in a rapid increase in the computational resources required for the optimal map search described in the previous section. Therefore, the size of M can be limited by keeping the number of edges in M ​​below kΓ. This can be achieved by choosing to keep kΓ heaviest edges while discarding the rest if M determined at any stage has more than kΓ edges, where k>1.

[0118] B.2 Genetic Algorithm

[0119] Figure 9 A flowchart of a method 900 for computing the Most Compact Cumulative Hypergraph (MCCS) using a genetic algorithm is depicted. The genetic optimization algorithm stores a population of subgraphs in its memory. In each generation, a subset of the population is selected to proceed to the next generation based on their health. Additionally, mutated copies of the healthiest members of the population may be generated. During mutation, the structure of the graph undergoes random changes. Some parent pairs can be selected from the current population and used in a crossover function to produce the next generation of children. These children are expected to combine and improve upon the good traits of their parents. A small subset is selected for mutation, where the structure of the graph undergoes random changes. The population size remains constant from generation to generation. After a certain number of generations, the best member of the surviving population is returned as the optimal solution.

[0120] The distortion ε(G) shown in the minimization formula (2) is used to calculate the distortion. c G s In the method 900 for the most compact cumulative hypergraph (MCCS), the distortion ε(G) shown in equation (2) is... c G s The negative value of is considered a health function in the genetic algorithm. In some embodiments, in order to compute the candidate graph G for the system graph G, s This level of health, searching for each circuit diagram G c The optimal vertex mapping to the candidate graph G is determined. It should be noted that computing the health function is already a challenging task, as it requires calculating the optimal mapping from the circuit diagram to the system graph. To compute this mapping, the aforementioned backtracking heaviest cumulative subgraph (HCs) and maximum common edge subgraph (mCES) algorithms are used. Therefore, the performance of the genetic algorithm is closely related to the computational performance of the mapping.

[0121] In box 902, a batch of C quantum circuits to be calibrated is received via classical computer 102.

[0122] In box 904, based on a batch received by classical computer 102... Quantum circuits To calculate circuit diagram Gc =(V c E c The set of , w).

[0123] In box 906, the distortion ε(G) shown in equation (2) is calculated by classical computer 102. c G s The health function with negative values.

[0124] In block 908, some candidate graphs G are mutated (i.e., some circuit graphs G that are not in the candidate graphs G are used). c Randomly replace some candidate graphs G). At a mutation rate p... m Make graph G = (V, E) abruptly (where the number of edges e ∈ E equals the calibration budget Γ) include p from all possible edges in candidate graph G. m |E| new edges, and randomly sample (1-p) edges from the original graph G, e∈E. m )*|E| edges.

[0125] In block 910, a cross is performed on pairs of two candidate graphs G. The two graphs that produce the cross (i.e., parent graphs) are G1 = (V, E1) and G2 = (V, E2), and the other two graphs are G3 = (V, E3) and G4 = (V, E4), where edges E3 and E4 are generated by sampling from E1∪E2 with the constraint |E3| = |E4| = Γ. If the health function of candidate graph G is less than a predetermined threshold, method 900 returns to block 906. If the health function of candidate graph G exceeds the predetermined threshold, method 900 proceeds to block 912. In another embodiment, a predetermined number n can be considered. G The intersection returns to box 906n. G The number of times, of which method 900 is in n G After the next iteration, we proceed to box 912.

[0126] In box 912, the calculated healthiest candidate graph G contains the calibration set S as its edges and is output as a result of the calculation.

[0127] III.C. Example

[0128] The following section presents example simulation results for optimizing quantum gate calibration for a given calibration budget Γ. In the example disclosed herein, the quantum circuit executes on a quantum processor 106, which includes 11 trapped ions in an ion trap quantum computing system 100. The quantum circuit executes on 10 to 20 physical qubits.

[0129] To demonstrate the quality of the method described herein, example simulation results are compared with the following primitive calibration techniques. The first primitive mapping technique (referred to as the "random method") is a completely unoptimized, trivial method that calibrates Γ random gates and then executes the circuit without any mapping between logical and physical qubits. The second primitive mapping technique (referred to as the "naive method") is a slightly optimized but still naive method that calculates the number of gates in each circuit and calibrates Γ of the most frequently used gates without any mapping between logical and physical qubits. As described above, in block 822 of the MCCS-based method 800, the method uses data from the circuit diagram G... c To system diagram G s This is a round of (re)mapping. Applying this (re)mapping over random or naive methods is referred to as the respective, asterisked methods. It should be noted that these asterisked methods rely on the maximum common edge subgraph (MCEs) or heaviest cumulative subgraph (HCs) mapping functions described in this paper.

[0130] Figure 10A and Figure 10B Example simulation results depict the average fidelity of the input quantum circuit c as a function of the calibration budget Γ for 30 of the most common unweighted graphs, and the decrease in the required calibration budget Γ to achieve the target average fidelity. In the example shown, the associated distortion ∈ after calibration decreases. + For ∈ + =1%, the distortion associated with calibration ∈ - For ∈ - =10%. In Figure 10A In the diagram, dashed lines indicate the fidelity of naive and randomly generated system graphs. The blue area covers the 90% confidence interval for random assignments from best to worst. Solid lines show the fidelity of averaged random and naive methods using qubit mappings (* indicates the existence of such mappings), as well as the fidelity generated using the Most Compact Cumulative Hypergraph (MCCS) and genetic algorithms. The averaged random method is averaged over 100 random instances. Figure 10B The diagram illustrates a method for fully utilizing qubit mappings to reduce the Γ requirement, with the baseline being the naive method. The reduction is calculated from the value obtained by applying... Figure 10A Obtained by linear interpolation of data points (not shown) Figure 10A The difference between the expected Γ demand and the average fidelity curve in the curve.

[0131] The average fidelity of all methods (using the MCCS algorithm, genetic algorithm, and original technique) is assumed to be Γ calibration gates with distortion ∈ + =1%, while the rest have distortion ∈ _ It is estimated at 10%.

[0132] Figure 11A and Figure 11B Example simulation results are presented for the average fidelity of the input quantum circuit c as a function of the calibration budget Γ for 40 of the most common weighted graphs, and for the reduction in the required calibration budget Γ to achieve the target average fidelity. In the example shown, the associated distortion reduction after calibration is ∈ + For ∈ + =1%, the distortion associated with calibration ∈ - For ∈ - =10%. In Figure 11A In the diagram, the dashed lines show the fidelity of naive and randomly generated system graphs without individual circuit remapping. The blue area covers the 90% confidence interval for random assignments from best to worst. The solid lines show the fidelity of the averaged random method and the naive method using qubit mapping (* indicates the existence of the mapping), as well as the fidelity generated using the Most Compact Cumulative Hypergraph (MCCS) and a genetic algorithm. The averaged random method is averaged over 100 random instances. Figure 11B The diagram illustrates a method for fully utilizing qubit mappings to reduce the Γ requirement, with the baseline being the naive method. The reduction is calculated from the Γ requirement obtained by applying the Γ method to the qubit mappings. Figure 11A Obtained by linear interpolation of data points (not shown) Figure 11A The difference between the expected Γ demand and the average fidelity curve in the curve.

[0133] Figure 12A and Figure 12B Example simulation results are presented to illustrate the reduction in the number of calls to the search function and the requirement for the calibration budget Γ to achieve the target average fidelity within the beam search version of the MCCS algorithm, compared to the naive method which assigns the same fidelity for different system sizes N and Γ = N, 2N, 3N. Figure 12A The figure shows the number of planning calls to the search function in the optimal random method using mapping for N = 10, 12, 14 and Γ = 2N. The error bars mark the 90% confidence interval of the random distribution. Figure 12B The decrease in N for each N is calculated as the additional Γ required to achieve the same fidelity as the MCCS method using the naive method. The average fidelity curve of the naive method is obtained by linear interpolation of the data points (not shown). Figure 12B The line in the diagram shows the decrease Γ as a function of the system size N. reduced The health function is of the form Γ. reduced =Γ(N)+C x (N-10), where C is the number of cases. x They were estimated to be 1.5, 2.1 and 1.9 respectively.

[0134] The scalability of the method described herein has been demonstrated in examples using artificially generated batch circuits for different system sizes, each batch consisting of five linear graphs, five star graphs, five random trees, and N randomly selected small-diameter regular graphs, where N is the number of qubits. This particular composition is inspired by linear, star, and tree graphs that frequently appear in quantum programs running on quantum computing systems such as the ion trap quantum computing system 100. Small-diameter regular graphs of the form (k, d, n) are selected, where k is the degree of the graph, d is the diameter of the graph, and n is the number of vertices, as they are customized to fully utilize the full-pair full-qubit connectivity available in quantum computing systems such as the ion trap quantum computing system 100. In the example shown, MCCS solutions were computed for batches generated from six different system sizes of N = 10..20 qubits, with calibration budgets Γ = N, 2N, 3N considered. Figure 12A The number of calls to the search function is shown as a measure of execution time. The expected number of calls for the optimal random method is also calculated by first calculating the optimal average fidelity as a function of the number of random trials for the baseline circuit, and then extrapolating it to match the average fidelity obtained by the MCCS method. This is then converted to the number of calls to the search function, in the form M / 2 × (the number of search calls for the MCCS solution), according to the method described herein. For solutions F = 2N, the expected number of trials is M ≈ 700 for N = 10, M ≈ 800 for N = 12, and M ≈ 750 for N = 14. Figure 12A This is more than 2.5 orders of magnitude faster than the MCCS algorithm in execution time. The actual time may be much longer because the upper limit of the 90% confidence interval spans five orders of magnitude due to the Poisson distribution of the probability of finding a solution randomly. For each case Γ = N, 2N, 3N considered in this paper, the associated distortion ∈ [value missing] was estimated using the previously used [value missing]. ± The expected average fidelity from the baseline batch of circuits. It has been found that if the naive method is used, more than twice the number of calibration gates are required to achieve the same average fidelity. According to Figure 12B As shown in the trend, the advantage of the Γ requirement increases with system size. It should be noted that, to achieve the same fidelity, for a 20-qubit system, the naive method may require up to 80 additional calibrations on top of the 60 already required by the better-performing MCCS algorithm.

[0135] III.D. Discussion

[0136] In the embodiments described herein, a method for minimizing calibration resources for a batch of quantum circuits is provided. It has been shown that, for both unweighted and weighted graphs, an optimized mapping between logical and physical qubits described by the graph can significantly impact fidelity. In both cases, the computational mapping method described herein provides an average increase in algorithmic fidelity from approximately 70% to over 90% compared to unmapped execution. Calibrating the gate set using a backtracking algorithm consistently demonstrates better performance for both weighted and unweighted graphs within the high fidelity range of interest compared to the naive method used to calibrate the most commonly used gates. In both cases, the genetic algorithm performs well for large calibration budgets Γ > 15. It should be noted that, with the aid of the remapping algorithm described herein, even randomized calibration sets can yield performance close to that obtained using backtracking or genetic algorithms. It should also be noted that random sampling methods require several orders of magnitude more time than backtracking algorithms to reach the target fidelity.

[0137] When using genetic algorithms to find the optimal calibration sequence, it was found that for large target budgets Γ, genetic algorithms outperform the MCCS algorithm in terms of average fidelity. However, it should be noted that genetic algorithms, which rely on the stochastic evolution of a series of candidate solutions, often consume significant time and computational resources. Other neural network-based methods, such as deep learning, can also be used to reduce the time required. More efficient integration with other search methods (such as backtracking or nested candidate methods) can further reduce computational resource requirements.

[0138] The method described in this paper can benefit quantum circuits in compiling and executing quantum algorithms, facilitating a wide range of numerical optimization problems. Furthermore, the method described in this paper allows smaller-scale quantum computers to optimize calibration routines for larger quantum computers.

[0139] It should be noted that the specific example embodiments described above are merely some possible examples of applying calibration resource optimization to quantum computing systems according to this disclosure, and do not limit the possible configurations, specifications, etc., of quantum computing systems according to this disclosure. For example, the quantum processor within a quantum computing system is not limited to a set of trapped ions having the aforementioned all-pair, fully connected nature. For example, the quantum processor can be an architecture with more stringent connectivity, such as superconducting qubits and modular topologies, where several strongly connected modules communicate with several channels. The graph theory techniques provided herein can be modified to reduce routing and shuttle times in such systems with limited connectivity.

[0140] While the foregoing relates to specific embodiments, other and further embodiments may be devised without departing from its basic scope, the scope of which is defined by the appended claims.

Claims

1. A method for performing a quantum computing process, the method comprising: By mapping multiple logical qubits of a classical computer to multiple physical qubits of a quantum processor, multiple quantum circuits can be executed using the physical qubits of the quantum processor, while minimizing the total distortion of the multiple quantum circuits. Each physical qubit includes a trapped ion, and Each of the plurality of quantum circuits includes a plurality of single-qubit gates and a plurality of two-qubit gates within the plurality of logical qubits; Select a first pair of physical qubits from the plurality of physical qubits, such that the first pair of physical qubits has maximum overlap with the plurality of quantum circuits; The system controller adjusts the amplitude and frequency of the laser pulse to be applied to each pair of physical qubits in the first plurality of pairs of physical qubits to correct the error in the biqubit gate in the first plurality of pairs of physical qubits, thereby reducing the distortion of the biqubit gate in the first plurality of pairs of physical qubits. The plurality of quantum circuits are executed on the quantum processor by applying laser pulses to the plurality of physical qubits, each laser pulse causing single-qubit gate operations and two-qubit gate operations in each of the plurality of quantum circuits; After the plurality of quantum circuits are executed on the quantum processor, the population of the qubit states of the physical qubits in the quantum processor is measured by the system controller; as well as The population of the qubit states of the physical qubits, measured by the output of the classical computer, is used as the execution result of the plurality of quantum circuits, wherein the execution result of the plurality of quantum circuits is configured to be displayed on a user interface, stored in the memory of the classical computer, or transmitted to another computing device.

2. The method according to claim 1, further comprising: Before executing the plurality of quantum circuits on the quantum processor, the population of the qubit states of the first plurality of pairs of physical qubits is measured by the system controller.

3. The method according to claim 1, further comprising: The classical computer calculates multiple circuit diagrams based on the multiple quantum circuits, each circuit diagram having multiple vertices representing the multiple logical qubits and multiple edges representing two-qubit gates between pairs of the multiple logical qubits.

4. The method according to claim 3, further comprising: The most compact cumulative hypergraph of the plurality of circuit diagrams is calculated using the classical computer. and The first multiple pairs of physical qubits are calculated by the classical computer based on the most compact cumulative hypergraph of the calculated multiple circuit diagrams.

5. The method according to claim 3, further comprising: The classical computer replaces one of the second plurality of circuit diagrams with one of the third plurality of circuit diagrams at a predetermined rate, wherein the plurality of circuit diagrams includes the second plurality of circuit diagrams and the third plurality of circuit diagrams. The classical computer generates a pair of circuit diagrams based on a pair of circuit diagrams in the second plurality of circuit diagrams and adds them to the second plurality of circuit diagrams. The most compact cumulative hypergraph of the plurality of circuit diagrams is calculated using the classical computer. and The first plurality of physical qubits are calculated by the classical computer based on the most compact cumulative hypergraph that includes the second plurality of circuit diagrams.

6. The method of claim 1, wherein the number of the first multiple pairs of physical qubits is less than a predetermined calibration budget.

7. A quantum computing system, comprising: A quantum processor comprising multiple physical qubits, each of which comprises a trapped ion; A classic computer is configured as follows: Multiple logical qubits are mapped to multiple physical qubits, such that multiple quantum circuits can be executed using the physical qubits, and the total distortion of the multiple quantum circuits is minimized, wherein each of the multiple quantum circuits includes multiple single-qubit gates and multiple two-qubit gates within the multiple logical qubits; Selecting a first plurality of pairs of physical qubits from the plurality of physical qubits, such that the first plurality of pairs of physical qubits have maximum overlap with the plurality of quantum circuits; and The system controller is configured as follows: The amplitude and frequency of the laser pulse to be applied to each pair of physical qubits in the first plurality of pairs of physical qubits are adjusted to correct the error in the biqubit gate in the first plurality of pairs of physical qubits, thereby reducing the distortion of the biqubit gate in the first plurality of pairs of physical qubits. The plurality of quantum circuits are executed on the quantum processor by applying laser pulses to the plurality of physical qubits, each laser pulse causing single-qubit and two-qubit gate operations in each of the plurality of quantum circuits, and After executing the plurality of quantum circuits on the quantum processor, the population of the qubit states of the physical qubits in the quantum processor is measured, wherein... The classic computer is also configured to: The measured population of the qubit states of the physical qubits is output as the execution result of the plurality of quantum circuits, wherein the execution result of the plurality of quantum circuits is configured to be displayed on a user interface, stored in the memory of the classical computer, or transmitted to another computing device.

8. The quantum computing system according to claim 7, wherein Before executing the plurality of quantum circuits on the quantum processor, the population of the qubit states of the first plurality of pairs of physical qubits is measured.

9. The quantum computing system of claim 7, wherein the classical computer is further configured to: Multiple circuit diagrams are calculated based on the multiple quantum circuits, each circuit diagram having multiple vertices representing the multiple logical qubits and multiple edges representing two-qubit gates between pairs of the multiple logical qubits.

10. The quantum computing system of claim 9, wherein the classical computer is further configured to: Calculate the most compact cumulative supergraph of the plurality of circuit diagrams; and The first multiple pairs of physical qubits are calculated based on the most compact cumulative hypergraph of the calculated multiple circuit diagrams.

11. The quantum computing system of claim 9, wherein the classical computer is further configured to: Replace one of the second plurality of circuit diagrams with one of the third plurality of circuit diagrams at a predetermined rate, wherein the plurality of circuit diagrams includes the second plurality of circuit diagrams and the third plurality of circuit diagrams; A pair of circuit diagrams are generated based on a pair of circuit diagrams in the second plurality of circuit diagrams and added to the second plurality of circuit diagrams; The most compact cumulative hypergraph of the plurality of circuit diagrams is calculated using the classical computer. and The first multiple pairs of physical qubits are calculated based on the most compact cumulative hypergraph that includes the second multiple circuit diagrams.

12. The quantum computing system of claim 7, wherein the number of the first multiple pairs of physical qubits is less than a predetermined calibration budget.

13. The quantum computing system according to claim 7, wherein Each of the trapped ions is an ion having both nuclear spin and electronic spin, with the difference between the nuclear spin and the electronic spin being zero.

14. The quantum computing system according to claim 13, wherein Each of the trapped ions is a nuclear spin and Ions in hyperfine states.

15. A quantum computing system comprising a non-volatile memory storing a plurality of instructions, which, when executed by one or more processors, cause the quantum computing system to perform operations, the operations including: A classical computer maps multiple logical qubits to multiple physical qubits of a quantum processor, enabling multiple quantum circuits to be executed using the physical qubits of the quantum processor, while minimizing the total distortion of the multiple quantum circuits. Each physical qubit includes a trapped ion, and Each of the plurality of quantum circuits includes a plurality of single-qubit gates and a plurality of two-qubit gates within the plurality of logical qubits; Select a first pair of physical qubits from the plurality of physical qubits, such that the first pair of physical qubits has maximum overlap with the plurality of quantum circuits; The system controller adjusts the amplitude and frequency of the laser pulse to be applied to each pair of physical qubits in the first plurality of pairs of physical qubits to correct the error in the biqubit gate in the first plurality of pairs of physical qubits, thereby reducing the distortion of the biqubit gate in the first plurality of pairs of physical qubits. The plurality of quantum circuits are executed on the quantum processor by applying laser pulses to the plurality of physical qubits, each laser pulse causing single-qubit gate operations and two-qubit gate operations in each of the plurality of quantum circuits; After the plurality of quantum circuits are executed on the quantum processor, the population of the qubit states of the physical qubits in the quantum processor is measured by the system controller; and The population of the qubit states of the physical qubits, measured by the output of the classical computer, is used as the execution result of the plurality of quantum circuits, wherein the execution result of the plurality of quantum circuits is configured to be displayed on a user interface, stored in the memory of the classical computer, or transmitted to another computing device.

16. The quantum computing system of claim 15, wherein the operation further comprises: Before executing the plurality of quantum circuits on the quantum processor, the population of the qubit states of the first plurality of pairs of physical qubits is measured by the system controller.

17. The quantum computing system of claim 15, wherein the operation further comprises: The classical computer calculates multiple circuit diagrams based on the multiple quantum circuits, each circuit diagram having multiple vertices representing the multiple logical qubits and multiple edges representing two-qubit gates between pairs of the multiple logical qubits.

18. The quantum computing system of claim 17, wherein the operation further comprises: The most compact cumulative hypergraph of the plurality of circuit diagrams is calculated using the classical computer. and The first multiple pairs of physical qubits are calculated by the classical computer based on the most compact cumulative hypergraph of the calculated multiple circuit diagrams.

19. The quantum computing system of claim 17, wherein the operation further comprises: The classical computer replaces one of the second plurality of circuit diagrams with one of the third plurality of circuit diagrams at a predetermined rate, wherein the plurality of circuit diagrams includes the second plurality of circuit diagrams and the third plurality of circuit diagrams. The classical computer generates a pair of circuit diagrams based on a pair of circuit diagrams in the second plurality of circuit diagrams and adds them to the second plurality of circuit diagrams. The most compact cumulative hypergraph of the multiple circuit diagrams is calculated using a classical computer. and The first plurality of physical qubits are calculated by the classical computer based on the most compact cumulative hypergraph that includes the second plurality of circuit diagrams.

20. The quantum computing system of claim 15, wherein the number of the first multiple pairs of physical qubits is less than a predetermined calibration budget.

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