Stochastic angle interpolation

Abstract

A method for implementing a rotation gate having a selectable rotation angle using a quantum device equipped with discretization gates, each having a discrete gate angle setting, the method comprising: (i) receiving an instruction to apply a rotation gate having a selected rotation angle; (ii) determining at least three discretization gates, each having a different discrete gate angle setting, based on the selected rotation angle; (iii) determining relative frequencies; (iv) selecting one of the determined discretization gates based on the determined relative frequencies; (v) applying the selected discretization gate to a qubit; (vi) measuring the state of the qubit and providing an output; (vii) repeating steps (iv) through (vi) multiple times; and (viii) combining the outputs from step (vi) to obtain a combined output based on the selected discretization gate.

Description

[Technical Field]

[0001] This invention relates to quantum computing. [Background technology]

[0002] In quantum algorithms, quantum gates are applied to qubits. Generally, it is desirable to be able to implement a continuous set of parameterized gates. Parameterized Pauli gates are a suitable set of gates. These gates encompass most of the gate sets developed for quantum technology, including single-qubit X, Y, or Z rotations and 2-qubit XX entungling gates.

[0003] Parameterized Pauli gates are associated with a specific rotation angle and are sometimes called rotation gates. When a rotation gate is applied to a qubit, the state of the qubit rotates around the axis of the Bloch sphere by the rotation angle.

[0004] By measuring the state of the device's qubits after applying an appropriate sequence of rotation gates, a binary output is obtained, and averaging this over many iterations yields an estimate of the expected value of the system's observables.

[0005] The desired rotation angle of a gate is based on experimental requirements. However, in practice, it is not always possible to apply a gate with a desired rotation angle because the classical control infrastructure requires the discretization of the gate angle. One class of highly relevant hardware is the electron spin qubit in a semiconductor device, where the gate is typically implemented with the help of a potential applied through electrodes. In efficiently scalable machines, it is highly desirable to limit the number of different control fields required, which in turn limits the number of possible gate rotations. Nevertheless, for universal quantum computing, it is essential to be able to realize any gate rotation as desired for any algorithm.

[0006] One solution is to apply a rotation gate with the rotation angle closest to the desired rotation angle. However, whenever the desired rotation angle does not match the available rotation angles, a mismatch occurs because the applied rotation gate is different from the desired rotation gate.

[0007] To mitigate the impact of the difference between the desired rotation angle and the available rotation angles, a quantum device with a large number of available rotation angles can be used. In this way, an available rotation angle closer to the desired rotation angle can be selected. However, a drawback of this solution is that the required control circuits are extensive and complex, increasing the manufacturing cost of the device. Furthermore, this solution does not eliminate the mismatch between the desired rotation angle and the available rotation angles. Moreover, the power consumption of quantum devices with a large number of available rotation angles can be relatively high, which is undesirable, especially when operating at low temperatures such as the cryogenic temperatures used in solid-state technology.

[0008] An alternative solution is to apply a sequence of gates instead of a single gate. Using a discrete set of parameterized Pauli gates with different rotation angles, a continuously parameterized gate can be constructed with only logarithmic overhead. However, the drawback of this solution is the increased circuit depth required to implement a particular rotation gate.

[0009] It is desirable to implement a rotary gate with a desired rotation angle without requiring deep circuitry or complex hardware. [Overview of the Initiative]

[0010] One aspect of the present invention provides a method for implementing a rotation gate having a selectable rotation angle using a quantum device comprising discretization gates, each having a discrete gate angle setting. The method includes (i) receiving an instruction to apply a rotation gate having a selected rotation angle, wherein the selected rotation angle is not the same as any of the discrete gate angle settings; (ii) determining at least three discretization gates, each having a different discrete gate angle setting, based on the selected rotation angle, wherein the first discrete gate angle setting is smaller than the selected rotation angle, the second discrete gate angle setting is larger than the selected rotation angle, and the third discrete gate angle setting differs from the selected rotation angle by an angle greater than π / 2; and (iii) determining a relative frequency for selecting each of the at least three determined discretization gates. The method further includes (iv) selecting one of the at least three determined discretization gates based on the determined relative frequency; (v) applying the selected discretization gate to a qubit; and (vi) measuring the state of the qubit and providing an output. The method further includes (vii) repeating steps (iv) through (vi) multiple times, and (viii) combining the outputs from step (vi) to obtain a combined output based on a selected discretization gate.

[0011] The advantage of this method is that continuous rotation gates can be implemented using low-complexity quantum devices with shallow circuits and low power consumption. The circuits are shallow because the step of applying a rotation gate with a selected rotation angle to a qubit is replaced, rather than added, by the application of one of the determined discretization gates at each circuit iteration. Discretizing the rotation angle of the quantum gates favorably reduces device complexity, lowers manufacturing costs, and provides lower power consumption. By determining the relative frequency for selecting each of at least three determined discretization gates and applying each of the discretization gates accordingly, it becomes advantageously possible to obtain a coupled output corresponding to the average output of rotation gates with a selected rotation angle.

[0012] Various hardware platforms that can utilize this method, including semiconductor electron spin devices, typically require extremely low temperatures, such as below 4 Kelvin, and therefore, low power consumption is a significant advantage.

[0013] The terms "gate" or "rotational gate" are sometimes also called rotation, gate operation, or rotational operation. These terms typically refer to the superoperator representation of a parameterized Pauli gate, i.e.,

number

[0014] The selectable rotation angles may preferably be rotation angles that can be selected from any rotation angle around the Bloch sphere. Optionally, the selectable rotation angles may be rotation angles that can be selected from one or more portions of the Bloch sphere. Optionally, the selectable rotation angles may be selected with finite precision.

[0015] The quantum device has discretized gates, and each discretized gate has a discrete gate angle setting. The quantum device having discretized gates may be a quantum device configured to execute a set of discretized gates each having a different discrete gate angle setting. In practice, many types of quantum devices must have discretized gates because it is impossible to realize a set of continuous gates without using a significant additional quantum resource. Each gate corresponds to a specific rotation around the axis of the Bloch sphere. The rotation angle associated with a particular discretized gate is fixed.

[0016] The relative frequency is preferably determined to enable the combined output to correspond to the average output of the rotation gate having the selected rotation angle.

[0017] When at least three discretized gates are determined based on the selected rotation angle and the relative frequencies for selecting these gates are determined, one of the at least three determined discretized gates is selected. The selection is performed based on the determined relative frequencies, but the order in which the discretized gates are applied in a continuous repetition process is not important and does not affect the combined output.

[0018] For each repetition, the selected discretized gate is applied to the qubit. The qubit is typically initialized prior to the application of the discretized gate. For each repetition, the qubit may be the same qubit or a different qubit.

[0019] Typically, quantum algorithms require circuits with multiple parameterized gates. Preferably, when a circuit with multiple parameterized gates is executed, the above method is applied individually to each quantum gate. Preferably, a method is provided for implementing first and second rotation gates having first and second selectable rotation angles using a quantum device equipped with discretization gates each having discrete gate angle settings, the method comprising: (i) receiving an instruction to apply first and second rotation gates each having first and second selected rotation angles, wherein the first and second selected rotation angles are not the same as any of the discrete gate angle settings; (ii) (a) determining at least three discretization gates of a first set, each having different discrete gate angle settings, wherein the first first discrete gate angle setting is smaller than the first selected rotation angle, the first second discrete gate angle setting is larger than the first selected rotation angle, and the first third discrete gate angle setting differs from the first selected rotation angle by an angle greater than π / 2; and (ii) (b) determining different discrete gates of a first set, each having different discrete gate angle settings, based on the second selected rotation angle. (iii)(a) a step of determining at least three discretization gates of a second set having discrete gate angle settings, wherein at least three discretization gates of the first set and the second set may include one or more of the same discretization gates, the second first discrete gate angle setting of the second is smaller than the second selected rotation angle, the second second discrete gate angle setting of the second is larger than the second selected rotation angle, and the second third discrete gate angle setting differs from the second selected rotation angle by an angle greater than π / 2; (iii)(a) a step of determining a first relative frequency for selecting each of the at least three determined discretization gates in the first set; (iii)(b) a step of determining a second relative frequency for selecting each of the at least three determined discretization gates in the second set; (iv)(a) a step of selecting one of the at least three determined discretization gates from the first set as the first selected discretization gate based on the first determined relative frequency;(iv)(b) Based on the second determined relative frequency, selecting, as the second selected discretized gate, one of at least three determined discretized gates from the second set; (v) applying the first and second selected discretized gates to the qubit; (vi) measuring the state of the qubit to provide an output; (vii) repeating steps (iv) to (vi) a plurality of times; (viii) combining the outputs from step (vi) to obtain a combined output based on the selected discretized gates.

[0020] Optionally, the first and second rotation angles are different from each other. However, the first and second rotation angles may be the same. If the first and second rotation angles are the same, the first and second sets are the same, and the first and second relative frequencies are the same.

[0021] Advantageously, this method implements a sequence of rotation gates with selectable rotation angles using a shallow circuit and a low-complexity quantum device with low power consumption. The measurement overhead typically scales, in the worst-case scenario, according to

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[0022] Typically, the method involves receiving an instruction to apply a sequence of N rotation gates, where N > 2, and each of the N rotation gates has a selected rotation angle, which may be the same or different. Preferably, based on the selected rotation angle of each of the N rotation gates, at least three discretization gates of N sets are determined. Two or more of the N sets may be the same. Preferably, N relative frequencies are determined to select each of the at least three determined discretization gates in each of the N sets. Preferably, based on the corresponding determined relative frequencies, one of the at least three determined discretization gates is selected from each of the N sets, thereby obtaining N selected discretization gates. Preferably, the N selected discretization gates are applied sequentially to a qubit. Preferably, the step of measuring the state of the qubit to provide an output is performed after the N selected discretization gates have been applied. Preferably, the steps of selecting the N discretization gates, applying the gates sequentially, and measuring the state of the qubit to provide an output are performed multiple times. Preferably, the outputs from the measurements are combined based on a selected discretization gate to obtain a combined output.

[0023] If multiple discretization gates are applied to a qubit in step (v), the qubit is typically initialized prior to the application of the first discretization gate in the sequence for each iteration.

[0024] Optionally, the circuit to be executed may include thousands of parameterized gates, for example, up to 10,000 parameterized gates. Optionally, the circuit may include non-parameterized gates in addition to the parameterized gates. These non-parameterized gates may be selected from different families of gates, for example, CNOT gates, or one of the discretized rotation gates such that the desired rotation angle coincides with one of the discrete angles, for example, a Pauli X-bit flip. Such non-parameterized gates can be implemented directly without introducing measurement overhead and without using the methods described. Optionally, one or more quantum operations can be performed between step (v) and step (vi).

[0025] Discretization gates may be applied to one qubit or two qubits. Advantageously, this increases the flexibility of quantum devices to run different quantum algorithms, including single-qubit and two-qubit quantum gates.

[0026] For each iteration, the state of the qubit is finally measured after applying a selected discretization gate or multiple gates to the qubit. Optionally, this is done using a measurement device or a readout device. The output from the measurement is typically binary, i.e., the state of the qubit.

number

number

[0027] The steps of selecting a gate or a set of gates, applying the gate to a qubit, and measuring the state of the qubit are performed multiple times. For each iteration, several quantum operations are typically performed before measuring the qubit state. Preferably, these steps are repeated at least 100 times, although some applications may allow fewer iterations. After multiple iterations, the "0" or "1" measurements are combined to obtain a combined output. The combined output is preferably determined by multiplying each measured output by +1 or -1 according to the selected discretization gate or set of gates, adding the outputs together, and scaling the sum according to the total number of measurements and the global prefactor or norm. The number of measurements is typically scaled according to the number of iterations.

[0028] The combined output typically corresponds to the average output of a rotation gate with a selected rotation angle. Typically, if a quantum device were configured to perform rotation gates, the average output of a rotation gate would be obtained by applying a rotation gate with a selected rotation angle to a qubit, measuring the state of the qubit to provide an output (which is binary), and repeating these steps multiple times. It should be noted that the average output is a theoretical construct because it is impossible to apply a rotation gate with a selected rotation angle using a quantum device with discretized gates, and in the relevant hardware implementation, it is impossible to have a physical quantum device with continuously parameterized gates. Therefore, the average output is preferably the theoretical average output that would have been obtained if it had been possible to apply a rotation gate with a selected rotation angle. The advantage of this method is that a physical quantum device with discretized gates effectively has the ability to implement continuously parameterized gates.

[0029] Preferably, the selection in step (iv) is performed randomly based on the determined relative frequencies. For example, step (iii) may include determining the probability of selecting each of the at least three determined discretization gates, and step (iv) may include selecting one of the at least three determined discretization gates based on the determined probabilities.

[0030] Such random selection simplifies the control infrastructure advantageously because it requires less hardware to implement. Alternatively, it is possible to determine the order in which to execute at least three determined discretization gates and then select one of the at least three determined discretization gates based on the determined relative frequencies and the determined order.

[0031] Preferably, step (iii) includes expressing the rotation gate as a linear combination of the discretization gates determined from step (ii), and determining the coefficients in the linear combination for each of the determined discretization gates, wherein the coefficient for one of the determined discretization gates is negative. Preferably, if three discretization gates are determined, only one coefficient is negative. If more than three discretization gates are determined, one or more coefficients may be negative, but not all of them.

[0032] Advantageously, the formula can be used to determine the relative frequency for selecting each of the three determined discretization gates.

[0033] Furthermore, the sign of the coefficient can be advantageously used to determine how the measurements are combined in step (viii). Preferably, if the selected discretization gate has a positive coefficient in the formula, the output from the measurement of the qubit state when the gate is applied is multiplied by +1. Preferably, if the selected discretization gate has a negative coefficient in the formula, the output from the measurement of the qubit state when the gate is applied is multiplied by -1. Thus, if the coefficient is positive and the measurement output is recorded as "+1", its contribution to the sum is indeed +1, and if the coefficient is negative and the measurement output is recorded as "+1", its contribution to the sum is negative, i.e., -1. The opposite is true if the measurement is recorded as "-1". The combination of negative and positive outputs favorably replicates a pure quantum state.

[0034] Preferably, if multiple discretization gates are applied to a qubit, each measurement output is recorded along with the sequence of selected discretization gates applied to the qubit before the measurement. Typically, if the selected discretization gates for a particular iteration of steps (iv) through (vi) include an odd number of selected discretization gates with negative coefficients in the formula, the output from the measurement of the qubit state is multiplied by -1. Typically, if the selected discretization gates for a particular iteration of steps (iv) through (vi) include an even number of selected discretization gates with negative coefficients in the formula, the output from the measurement of the qubit state is multiplied by +1.

[0035] The first discrete gate angle setting is smaller than the selected rotation angle, and the second discrete gate angle setting is larger than the selected rotation angle. The first and second discrete gate angle settings correspond to the first and second discretization gates determined in step (ii).

[0036] Advantageously, this makes it possible to apply the ratio of under-rotated gates and over-rotated gates to qubits.

[0037] Preferably, the first and second discrete gate angle settings are the two discrete gate angle settings closest to the selected rotation angle.

[0038] Advantageously, this provides an efficient method for applying under-rotated and over-rotated gate ratios to qubits. Furthermore, by selecting the nearest discrete gate angle setting, measurement overhead is reduced compared to selecting a discrete gate angle setting further away from the selected rotation angle.

[0039] The third discrete gate angle setting differs from the selected rotation angle by an angle greater than π / 2. The third discrete gate angle setting corresponds to the third discretized gate determined in step (ii). This means that the third discrete gate angle setting is substantially opposite to the preferably selected rotation angle. This favorably reduces the variance of the coupled output obtained in step (viii).

[0040] Preferably, when the rotation gate is expressed as a linear combination of determined discretization gates, a third discretization gate having a discrete gate angle setting of π radians or 180 degrees, which is polar opposite to the selected rotation angle, has a negative coefficient. Preferably, when the third discretization gate is applied to a qubit, the sign of the output is inverted. Inverting the sign of the output means multiplying the output by -1. This is advantageous in restoring the pure quantum state.

[0041] Preferably, when a rotation gate is expressed as a linear combination of determined discretization gates, the sign of the output is inverted when a gate with a negative coefficient is applied to a qubit.

[0042] Preferably, first, second, and third discretization gates are determined, each having first, second, and third discrete gate angle settings, respectively. The first and second discrete gate angle settings may be the discrete gate angle settings closest to the selected rotation angle, while the third discrete gate angle setting may be polar, i.e., about π radians away from the selected rotation angle. Advantageously, the use of these three discretization gates provides the most efficient gate combination, requiring small overhead and providing a coupled output with low dispersion.

[0043] Optionally, additional discretization gates with additional different discrete gate angle settings are determined. Preferably, in this case, at least two of the determined discretization gates are within ±π / 2 radians of the selected rotation angle, and at least one of the determined discretization gates is beyond π / 2 radians from the selected rotation angle. Preferably, the output measured when a determined discretization gate within π / 2 radians of the selected rotation is applied to a qubit is multiplied by +1, and the output measured when a determined discretization gate beyond π / 2 radians from the selected rotation is applied to a qubit is multiplied by -1. Preferably, the difference between two of the discrete gate angle settings is about π. Advantageously, this ensures that two of the discrete gate angle settings are on opposite sides of the unit circle, helping to recover a pure or ideal quantum state. Typically, at least two discretization gates are within π / 2 radians of the selected rotation angle, and for example, they may be the two closest gate angle settings. Using these discretization gates that do not have gate angle settings on opposite sides of the unit circle may result in a mixed combined output, i.e., a qubit state inside the Bloch sphere. Applying two discretization gates with an angle difference of π or about π helps to recover an ideal quantum state for the combined output. This can be achieved by changing the sign of the output if the applied rotation angle of the selected discretization gates is opposite to the selected rotation angle. Optionally, if discretization gates with a rotation angle difference of π are not available in the quantum device, the difference between two of the discrete gate angle settings may be between π / 2 and 3π / 2.

[0044] Optionally, the discrete gate angle settings are uniformly distributed in the range of 0 to 2π. Alternatively, the discrete gate angle settings are unevenly distributed in the range of 0 to 2π. That is, the angular spacing between adjacent discrete gate angle settings may be constant or not. This can be adapted based on experimental requirements. For example, if the rotation gates implemented using a quantum device have a broad and relatively uniform angular distribution, it is preferable that the angular spacing between adjacent discrete gate angle settings is substantially similar. Alternatively, if the rotation gates implemented using a quantum device are unevenly distributed, for example, concentrated around a particular angular region, it is preferable that the angular spacing between adjacent discrete gate angle settings is similarly uneven. For example, the discrete gate angle settings may be concentrated in the region where the rotation angles of the implemented rotation gates exist. Advantageously, a variety of rotation gates can be implemented efficiently using a quantum device with discretized gates having uniformly distributed discrete gate angle settings.

[0045] Preferably, the repeating step (vii) is 10 2 From round 10 8 Preferably 10 times 5 From round 10 6 The process is executed for several iterations. The output from each measurement in step (vi) is binary, and therefore, repeating the measurement more times can favorably improve the accuracy of the combined output. In some cases, hardware constraints may cause each selected discretization gate to be applied multiple times. Optionally, if N parameterized gates are implemented sequentially, M different sequences of the N selected discretization gates may be determined, and each of the M sequences may be applied multiple times. This may be computationally more efficient than determining a different sequence for each iteration. Since the measurement overhead is negligible, the advantage of this method is that the additional resource cost to implement it is negligible. Preferably, step (ii) includes determining three discretization gates, each having a different discrete gate angle setting, based on the selected rotation angle. More than three discretization gates may be determined, but determining only three discretization gates favorably increases the efficiency of this method.

[0046] Optionally, step (ii) is to search a dataset for a selected rotation angle, wherein the dataset contains selectable rotation angles and at least three corresponding discretization gates for each selectable rotation angle, and to determine at least three discretization gates based on the at least three discretization gates in the dataset corresponding to the selected rotation angle. Advantageously, searching for suitable discretization gates in the dataset is an efficient way to determine at least three discretization gates to apply.

[0047] Optionally, the dataset further includes relative frequencies for selecting each of at least three discretization gates corresponding to each selectable rotation angle, and step (iii) includes searching the dataset for the selected rotation angle and determining relative frequencies for selecting each of the at least three determined discretization gates based on the relative frequencies in the dataset corresponding to the selected rotation angle. Advantageously, searching for relevant relative frequencies in the dataset is efficient.

[0048] Optionally, a dataset may be created following decision steps (ii) and / or (iii) each time that at least three discretization gates and relative frequencies are determined for a particular selected rotation angle. Optionally, after step (ii), the determined at least three discretization gates are stored in the dataset. Optionally, after step (iii), the determined relative frequencies corresponding to at least three discretization gates are stored in the dataset. Advantageously, the decision steps can be simplified because previously determined values ​​can be reused if an instruction to apply a previously applied rotation gate is received.

[0049] Preferably, steps (ii) through (iv) are computed before applying the selected gate or multiple gates in step (v). Optionally, steps (ii) through (iv) are computed for each of the multiple iterations before the first application of the selected gate or multiple gates. Advantageously, this reduces the execution time of the quantum algorithm.

[0050] Preferably, at least three determined discretization gates are 3 to 2 18 Determined from a set of discretization gates between individuals, preferably 2 12The results are determined from between 1,000 and 128 discretization gates, more preferably between 3 and 128. Advantageously, a smaller set of discretization gates, i.e., fewer discretization gates available to determine at least three subsets, requires less classical control infrastructure, less power to run, and less space occupied on the physical device. Furthermore, discretizing quantum gates advantageously reduces the complexity required to engineer precisely controlled quantum devices. Providing fewer discretization gates, i.e., fewer bits, further reduces the power consumption of the quantum device. Advantageously, increasing the number of discretization gates reduces the variance of the coupled output. The advantages of increasing and decreasing the number of discretization gates are preferably balanced. Since the increased variance is usually small, it is typically more advantageous to decrease the number of discretization gates. However, a minimum number of discretization gates is required to implement this method.

[0051] Typically, the steps of this method are performed using either a quantum device or a classical controller. Typically, a quantum device is configured to perform steps (v), (vi), and (vii) of this method, and a classical controller is configured to perform steps (i), (ii), (iii), (iv), and (viii) of this method.

[0052] Typically, the combined output corresponds to the expected value of the observable properties of the system. Therefore, this method can be usefully used to measure the expected value of the observable properties of a system by selecting a specific rotation angle corresponding to the observable quantity.

[0053] Typically, a quantum algorithm is executed using multiple qubits. For example, this method may include receiving instructions to apply rotation gates to each qubit of a plurality of qubits. The rotation gates applied to different qubits may be the same or different, i.e., they may have the same angle rotation or different angle rotations. Multiple rotation gates may be applied to each qubit. Preferably, this method is applied to each qubit individually, and the execution of steps (ii) to (viii) of this method for the first qubit does not affect the execution of steps (ii) to (viii) of this method for a second qubit different from the first qubit.

[0054] Another aspect of the present invention provides a quantum device comprising discretized gates each having a discrete gate angle setting. The quantum device is configured to execute the method according to the first aspect.

[0055] An advantage of the quantum device according to this aspect is that the power consumption is low due to the discretization of the gates. This has the further advantage that the low power consumption makes the device suitable for use at very low temperatures.

[0056] Optionally, the device is a semiconductor device comprising quantum dots configured to electrostatically trap electron spins. Advantageously, such a device is particularly suitable for implementing this method.

[0057] Optionally, the discretized gates are determined by a discrete set of possible voltage settings. Advantageously, this control mechanism is easy to implement.

[0058] Preferably, the quantum device has between 3 and 2 18 discretized gates, more preferably between 3 and 2 12 discretized gates, even more preferably between 3 and 2 7It has 128 discretization gates. Advantageously, quantum devices with fewer discretization gates consume less power and occupy less space due to reduced classical control infrastructure requirements.

[0059] Typically, quantum devices are silicon-based and may be fabricated using complementary metal-oxide-semiconductor fabrication processes.

[0060] One aspect of the present invention provides a method for implementing a rotation gate having a selectable rotation angle using a quantum device equipped with discretization gates, each having a discrete gate angle setting. The method includes (i) receiving an instruction to apply a rotation gate having a selected rotation angle, wherein the rotation angle is not the same as any of the discrete gate angle settings; (ii) determining at least three discretization gates, each having a different discrete gate angle setting, based on the selected rotation angle; and (iii) determining a relative frequency for selecting each of the at least three determined discretization gates. The method further includes (iv) selecting one of the at least three determined discretization gates based on the determined relative frequency; (v) applying the selected discretization gate to a qubit; and (vi) measuring the state of the qubit and providing an output. The method further includes (vii) repeating steps (iv) through (vi) multiple times; and (viii) combining the outputs from step (vi) to obtain a combined output based on the selected discretization gate. [Brief explanation of the drawing]

[0061] Embodiments of the present invention will be described below with reference to the accompanying drawings.

[0062] [Figure 1A] Figure 1A is a schematic diagram of a method for implementing a rotary gate. [Figure 1B] Figure 1B is a schematic diagram of a method for implementing a rotary gate according to one embodiment of the present invention. [Figure 2] Figure 2 is a graph showing the probability distribution. [Figure 3] Figure 3 is a graph showing the minimization of gradient descent. [Figure 4] Figure 4 is a schematic diagram of the gate angle setting. [Modes for carrying out the invention]

[0063] The ability to realize a continuous set of parameterized gates is required by many near-future quantum algorithms. A particularly useful set of gates is the so-called parameterized Pauli gates, which are suitable for representing physical platforms. Figure 1A schematically shows a theoretical quantum device 101 with a continuous gate angle setting. Such a device can be used to apply a rotation gate 102 having a selectable rotation angle 103. The rotation angle 103 is a continuous angle setting in Figure 1A. This means that the selectable rotation angle 103 in this example is any specified rotation around the rotation axis of the Bloch sphere, such as the X, Y, or Z axis. In this theoretical quantum device 101 with a continuous gate angle setting, due to the continuous nature of the gate angle setting, there is no limit to the precision of the selectable rotation angle. In an alternative example, the selectable rotation angle may be selectable from one or more distinct angular domains.

[0064] After applying a rotation gate 102 with a selected rotation angle 103 to a qubit, the measurement of the qubit's state returns a binary output, i.e., the qubit is in state |0> or |1>, which may be stored for subsequent processing by recording "+1" or "-1", respectively. The average output 106 is obtained by calculating the scaled average of the measurements. For example, if the objective is to estimate the expectation value of the Pauli Z operator of a qubit, and 400 out of 1000 measurements are recorded as "-1" and 600 out of 1000 measurements are recorded as "+1", then the average output 106 is ((400 × -1) + (600 × 1)) / 1000 = 0.2. This provides the expectation value of the observables of the quantum system. Instead of assigning numerical values ​​to the results |0> and |1>, other mathematical objects such as probability distributions, matrices, or so-called classical snapshots may be assigned to the individual outputs, which are also averaged.

[0065] However, in quantum computing technology, gate instructions (e.g., instructions to execute a specific rotation gate) must be scheduled by a classical control infrastructure. The user must specify the type of gate to implement and which qubits in the system to implement it on. The classical control infrastructure is typically digital, and therefore the gate angle must be discretized into B bits. That is, it is desirable to implement a rotation gate with a gate angle setting selected from a continuous angle range, but in many hardware implementations, it is not possible to do this in practice by providing a quantum device 101 with a continuous gate angle setting.

[0066] Embodiments of the present invention enable the implementation of a rotation gate 102 having a selectable rotation angle 103 using a quantum device having discretization gates, as shown in Figure 1B. In Figure 1B, the quantum device 111 having discrete gate angle settings has first, second, and third gates 112, 113, and 114 determined from a set of 10 discretization gates. Each determined discretization gate 112 to 114 has its respective discrete gate angle settings 122, 123, and 124, as shown in Figure 1B. The quantum device 111 having discrete gate angle settings can implement a rotation gate 102 having a selectable rotation angle 103, as shown in Figure 1A, using the method described herein.

[0067] The first, second, and third discrete gate angle settings 122 to 124 are three distinct angles. The remaining discretization gates (not shown) have discrete gate angle settings that are distinct from each other and also distinct from the first, second, and third discrete gate angle settings. In this example, the quantum device has three determined discretization gates 112 to 114 determined from a set of 10 discretization gates. However, in other examples, the quantum device 111 with discrete gate angle settings may have more determined discretization gates, and / or the set of discretization gates may be larger or smaller. Each gate is a rotation gate corresponding to a specific angular rotation.

[0068] In this method, the determined relative frequency 115 is used to select one of the determined discretization gates 112 to 114. The determined relative frequency 115 may be, for example, 50:49:1. This relative frequency indicates that in every 100 iterations, the first gate 112 should be applied 50 times, the second gate 113 49 times, and the third gate 114 1 time. The order in which the discretization gates 112 to 114 are applied is not considered important. In this example, the discretization gates 112 to 114 are selected randomly. The control infrastructure is randomly instructed to apply one of the determined discretization gates 112 to 114. Alternatively, the discretization gates 112 to 114 may be selected in an ordered manner, such as in blocks of the same gate, alternating between the determined gates, or in other predetermined sequences.

[0069] After applying the first, second, or third gates 112 to 114, each having a first, second, or third gate angle setting 122 to 124, to a qubit, the measurement of the qubit's state returns a binary output, i.e., the qubit is in state |0> or |1>, and classical memory records "+1" or "-1", respectively. The combined output 116 is obtained by adding the outputs, multiplying each output by +1 or -1 based on the selected discretization gate, and scaling the sum according to the total number of measurements and the global prefactor. In this example, the applied discretization gates are stored with the output for each iteration to determine the multiplication coefficient for each output, i.e., +1 or -1. In an example where a sequence of multiple discretization gates is applied for each iteration, the sequence of applied discretization gates is stored with the output for each iteration to determine the multiplication coefficient for each output.

[0070] The coupled output 116 obtained using the quantum device 111 with discrete gate angle settings corresponds to the theoretical average output 106 that would be obtained using the theoretical quantum device 101 with continuous gate angle settings.

[0071] The described method may be used when the selected rotation angle 103 is not the same as any of the available angles 122 to 124 in the device 111 having discrete gate angle settings. If the selected rotation angle 103 is equal to one of the discrete gate angle settings 122 to 124, the quantum device 111 having discrete gate angle settings is used to apply only the relevant discretized gate for 100% of the time for each iteration. The outputs from the measurements are then combined by averaging the measurements and multiplying the average by a coefficient, i.e., determining the scaled average.

[0072] When an instruction is received to apply a rotation gate 102 having a selected rotation angle 103, if the rotation angle 103 is not the same as any of the discrete angle settings 122 to 124, then at least three discretization gates 112 to 114 are determined based on the selected rotation angle 103. The three discretization gates 112 to 114 each have different discrete gate angle settings 122 to 124. A relative frequency 115 is determined for selecting each of the at least three determined discretization gates 112 to 114. Then, the first, second, or third discretization gate 112 to 114 selected based on the determined relative frequency 115, and the selected gates 112 to 114 are applied to the qubit. The state of the qubit after each discretization rotation gate 112 to 114 has been applied is finally measured to provide the output from each measurement, which is binary. Therefore, this method allows us to translate instructions to a quantum device into randomized instructions to a device that can only execute quantum gates with a discrete set of rotation angles, so that the combined output 116 is the same as the theoretical average output 106.

[0073] In this example, the selection of one of the determined discretization gates 112 to 114 is random. However, due to the determined relative frequencies, the exact, desired unitary rotation gate 102 with the selected rotation angle 103 is executed on average.

[0074] In this example, an increased number of circuit iterations is used to measure the combined output, i.e., the expected value of the observable. However, the measurement overhead resulting from the additional circuit iterations is negligible. For example, a circuit with 1000 parameterized gates and a discretization with a B=7 bit resolution (i.e., a quantum device) requires 2 discretization gates to determine at least 3 discretization gates for implementing a rotation gate with a selectable rotation angle. 7 In the case of having 128 discretization gates, the worst-case measurement overhead is approximately double. Aside from the slightly increased number of circuit iterations, it should be noted that an advantage of the described method is that no additional quantum resources are required.

[0075] In another example, an instruction is received to apply first and second rotation gates having first and second selected rotation angles. The first and second rotation angles are not the same as any of the discrete angle settings. Based on the first selected rotation angle, at least three discretization gates of a first set are determined. The first set includes the first, second, and third discretization gates. Based on the second selected rotation angle, at least three discretization gates of a second set are determined. The second set includes the second discretization gate as well as the fourth and fifth discretization gates. First and second relative frequencies are determined, respectively, to select each of the determined discretization gates in the first and second sets. The state of the qubit after each sequence of discretized rotation gates has been applied is measured to provide an output. Each sequence of applied rotation gates includes one discretization gate from the first set and one discretization gate from the second set.

[0076] Figure 2 shows the average output of a rotation gate with a selected rotation angle, the corresponding combined output using the methods described in relation to Figures 1A and 1B, and the average output of a comparative example. Figure 2 shows the probability distribution for the three scenarios described. Each scenario involves a 6-qubit circuit with ν = 120 parameterized gates and 1000 iterations, or "shots," with the Pauli operator

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[0077] The first probability distribution 21 is theoretically determined based on a theoretical quantum device that has infinite resolution and can therefore execute gates at any rotation angle on a continuous spectrum. As mentioned above, such a device is not feasible in many hardware implementations, but the effectiveness of a physical quantum device can be determined using theoretical predictions. The scaled mean 24 of the first probability distribution 21, expected value

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[0078] The second probability distribution 22 is theoretically determined based on a quantum device that can perform only rotations with 7-bit precision. This is because the quantum device has 2 discrete gate angle settings, each with a different discrete gate angle setting. 7This means there are a number of discretized gates. In this example, when a command is received to apply a rotation gate with a rotation angle other than the available gate angle settings, the discretized gate with the closest gate angle setting is applied. The resulting probability distribution 22 is biased, i.e., its mean is shifted, due to the accumulation of consistent mismatches due to over-rotation and under-rotation. The scaled mean 25 of the second probability distribution 22 is approximately 0.12.

[0079] The third probability distribution 23 was used to determine the second probability distribution 22. 7 =This is an experimentally estimated histogram based on a quantum device with 128 discretization gates. The difference in this scenario is that when an instruction is received to apply a rotation gate with a selected rotation different from any of the discrete gate angle settings, three discretization gates are determined based on the selected rotation and applied according to relative frequencies determined to accurately implement the desired rotation angles on average. As can be seen from Figure 2, this method is the expected value of the first probability distribution 21

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[0080] In Figure 2, the first and third probability distributions 21 and 23 have the same expected value.

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[0081] Expected value of the observed quantity in the third probability distribution 23

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[0082] Figure 3 further illustrates the comparative performance of a theoretical quantum device with infinite resolution and a quantum device with finite resolution, with and without employing the described method, namely the stochastic angular interpolation technique. Figure 3 shows the energy minimization for the three scenarios described, specifically the gradient descent minimization of the energy of the 12-qubit spin ring problem. This type of problem is a typical application of early quantum computers, aiming to determine the quantum state that minimizes the energy of a quantum Hamiltonian. Typically, the spin problem Hamiltonian is used because it is relevant to binary optimization in fundamental physics and commercial applications. The energy distance from the exact ground state energy, ΔE, is determined using a circuit with 540 parameterized gates. For such a small, classically scalable problem concerning 12 qubits, this is a relatively deep circuit, but for larger problem instances involving more than 50 qubits, the number of parameterized gates required could be on the order of thousands. In this example, some of the 540 parameterized gates are single-qubit gates, and some of the 540 parameterized gates are two-qubit gates.

[0083] The first dataset 31 is determined theoretically. In the first dataset 31, the optimization assumes a theoretical quantum device with ideal properties: infinite rotational angular resolution.

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[0084] Therefore, as can be seen from Figure 3, using discretization gates in practical quantum algorithms, as in the second dataset 32, can lead to a large accumulation of consistent discrepancies. 7 This is especially true when the number of discretization gates in a quantum device is small, such as when the value is 128.

[0085] However, by using the described angular interpolation method, the ideal operation of a quantum device with infinite angular resolution can be achieved using a practical quantum device with finite angular resolution, as shown in the third dataset 33. The described approach outperforms a simple rounding approach that could be employed using a quantum device with finite angular resolution, using the same number of circuit iterations and the same number of discretization gates. Figure 4 is a schematic diagram of discrete gate angle settings on a Bloch sphere.

[0086] As mentioned above, parameterized Pauli gates encompass most of the gate sets developed for quantum technology, including single-qubit X, Y, or Z rotations and two-qubit XX entangling gates. Pauli gates can be any Pauli string.

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[0087] Figure 4 shows the parameterized Pauli gate.

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[0088] The rotation angle to be applied is selected according to the experimental requirements. In this example, the selectable rotation angle 401 may be any rotation angle with unlimited precision and range. In an alternative example, the rotation angle may be selected from a specified range of rotation angles and / or with a specified maximum precision.

[0089] In this example, the selected rotation angle 401 lies between the first discrete gate angle setting 402 and the second discrete gate angle setting 403. The first and second discrete angle settings 402, 403 are adjacent, and there are no available discretization gates with angular rotations between the first and second discrete angle settings 402, 403 in the quantum device. The angular interval between the first and second discrete angle settings 402, 403 is Δ.

[0090] The first discrete gate angle setting 402 is

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[0091] Using the hyperoperator representation, the first discretization gate is

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[0092] In a comparative example using a quantum device with discretization gates, when an instruction is received to apply a rotation gate with a selected rotation angle different from the available discrete gate angle settings, the selected rotation angle is simply rounded to the nearest available discrete gate angle setting. Then, a discretization gate with the gate angle setting closest to the selected rotation angle is applied to the qubit.

[0093] The drawback of this comparative example is that, by rounding the selected angular rotation value to the nearest available angle, all gate applications will be either over-rotated or under-rotated if the selected rotation is not the same as one of the available rotations. This can lead to a large, consistent accumulation of mismatches in the applied angular rotations. In the worst case, a consistent under-rotation or over-rotation of Δ / 2 occurs, where Δ is the interval between adjacent available angular rotation settings. This leads to a biased probability distribution, as shown in Figure 2. Because a biased probability distribution has a shifted mean, it does not return an accurate combined output.

[0094] In a further comparative example using a quantum device with a discretization gate, a selected rotation angle different from the available discrete gate angle settings.

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[0095] The drawback of this comparative example is that, on average, non-unitary operations are obtained. This results in a mixed quantum state that is represented as a point contained within the Bloch sphere. Using the method described in this comparative example, Δ 2 A biased, non-coherent mismatch of the order of magnitude is obtained.

[0096] In this example, the Pauli rotation gate (as a quantum channel) is represented as a linear combination of three discretization gates. In addition to the first and second discretization gates with first and second discrete gate angle settings 402, 403, a third discretization gate with a third discrete gate angle setting 404 is also used. The third discretization gate is

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[0097] The selected rotation angle 401 is

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[0098] These coefficients

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[0099] In this example, adjacent discrete gate angle settings are separated by a small, uniform interval Δ. Therefore, the asymptotic equation of the coefficients can be written using the relative position λ of the rotational excess θ between the two discrete gate angle settings and λ = θ / Δ as follows:

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[0100] Therefore, the first discrete gate angle setting 402,

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[0101] In this example, a sampling scheme is defined in which one of three selected discretization gates is applied to a qubit. To do this, three rotation operators are used.

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[0102] In this example, the interval Δ between each discrete gate angle setting is the same. However, in alternative examples, the intervals between discrete gate angle settings may be different. If the angle discretization is not uniform, i.e., if the angular intervals between adjacent discrete gate angle settings are not the same, then the above equation becomes:

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[0103] small probability

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[0104] As described in relation to Figure 2, the increased variance of the observable

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[0105] In the example where there are multiple quantum rotation gates N continuously executed within the circuit gate the upper limit of the circuit repetition overhead is given as follows.

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[0106] As will be appreciated, a method for implementing continuous rotation gates using a quantum device having discrete gates is provided. Such a device is low power and low complexity and has enhanced capabilities that enable recovery of the performance of an ideal quantum device having infinite rotation angle resolution and infinite repetition.

Claims

1. A method for implementing a rotation gate with a selectable rotation angle using a quantum device equipped with discretization gates, each having a discrete gate angle setting, (i) receiving a command to apply a rotation gate having a selected rotation angle, wherein the selected rotation angle is not the same as any of the discrete gate angle settings, (ii) A step of determining at least three discretization gates, each having a different discrete gate angle setting, based on the selected rotation angle, wherein the first discrete gate angle setting is smaller than the selected rotation angle, the second discrete gate angle setting is larger than the selected rotation angle, and the third discrete gate angle setting differs from the selected rotation angle by an angle greater than π / 2, (iii) The step of determining the relative frequency for selecting each of the at least three of the determined discretization gates, (iv) A step of selecting one of the at least three determined discretization gates based on the determined relative frequencies, (v) The step of applying the selected discretization gate to a qubit, (vi) A step of measuring the state of the qubit and providing an output, (vii) A step that repeats step (iv) to (vi) multiple times, (viiii) A step of combining the outputs from step (vi) in order to obtain a combined output based on the selected discretization gate, A method that includes this.

2. Step (iii) includes determining the probability of selecting each of the at least three of the aforementioned discretization gates, Step (iv) includes selecting one of the three determined discretization gates based on the determined probability. The method according to claim 1.

3. Step (iii) includes representing the rotation gate as a linear combination of the determined discretization gates, and determining the coefficients in the linear combination for each of the determined discretization gates, wherein the coefficient for at least one of the determined discretization gates is negative. The method according to claim 1 or 2.

4. The first and second discrete gate angle settings are the two discrete gate angle settings closest to the selected rotation angle. The method according to any one of claims 1 to 3.

5. The difference between two of the discrete gate angle settings is approximately π radians. The method according to any one of claims 1 to 4.

6. Step (ii) includes determining three discretization gates, each having a different discrete gate angle setting, based on the selected rotation angle. The method according to any one of claims 1 to 5.

7. Step (ii) is Searching a dataset for the selected rotation angle, wherein the dataset includes selectable rotation angles and at least three corresponding discretization gates for each selectable rotation angle. This includes determining at least three discretization gates based on at least three discretization gates in the dataset corresponding to the selected rotation angle, The method according to any one of claims 1 to 6.

8. The dataset further includes relative frequencies for selecting each of the at least three discretization gates corresponding to each selectable rotation angle, The above step (iii) is, Searching the dataset for the selected rotation angle, The process includes determining the relative frequency for selecting each of the at least three determined discretization gates based on the relative frequency in the dataset corresponding to the selected rotation angle, The method according to claim 7.

9. (i) receiving a command to apply first and second rotation gates having first and second selected rotation angles, respectively, wherein the first and second selected rotation angles are not the same as any of the discrete gate angle settings, (ii) (a) A step of determining at least three discretization gates of a first set, each having a different discrete gate angle setting, based on the first selected rotation angle, the first first discrete gate angle setting being smaller than the first selected rotation angle, the first second discrete gate angle setting being larger than the first selected rotation angle, and the first third discrete gate angle setting being different from the first selected rotation angle by an angle greater than π / 2, (ii) (b) A step of determining at least three discretization gates of a second set, each having a different discrete gate angle setting, based on the second selected rotation angle, wherein the at least three discretization gates of the first set and the second set may include one or more of the same discretization gates, the second first discrete gate angle setting being smaller than the second selected rotation angle, the second second discrete gate angle setting being larger than the second selected rotation angle, and the second third discrete gate angle setting being different from the second selected rotation angle by an angle greater than π / 2, (iii) (a) The step of determining a first relative frequency for selecting each of the three determined discretization gates in a first set, (iii) (b) Determining a second relative frequency for selecting each of the three determined discretization gates in the second set, (iv) (a) Based on the first determined relative frequencies, the step of selecting one of the first set of at least three determined discretization gates as the first selected discretization gate, (iv) (b) Based on the second determined relative frequencies, the step of selecting one of the three determined discretization gates from the second set as the second selected discretization gate, (v) the step of applying first and second selected discretization gates to a qubit, (vi) the step of measuring the state of the qubit and providing an output, (vii) A step that repeats step (iv) to (vi) multiple times, (viiii) a step of combining the outputs from step (vi) to obtain a combined output based on the selected discretization gate, The method according to any one of claims 1 to 8.

10. A first and / or second selected discretization gate is applied to two qubits. The method according to claim 9.

11. The discretization gate is determined by a discrete set of possible voltage settings. The method according to any one of claims 1 to 10.

12. At least three of the aforementioned determined discretization gates are 3 to 2 １８ The set of discretization gates between the individual elements is determined, preferably 3 to 2. １２ The discretization gates are determined from between 3 and 128, more preferably from between 3 and 128. The method according to any one of claims 1 to 10.

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