Frequency patterns for reducing parasitic interactions in qubit grids

Abstract

To reduce parasitic interactions among quantum bits.SOLUTION: Provided is a method, a system, and a device for operating a system of qubits. In one embodiment, the method includes: a step for causing a first qubit derived from first multiple qubits to operate, in a first qubit frequency derived from a first qubit frequency area; and a step for causing a second qubit derived from the first multiple qubits to operate, in a second qubit frequency derived from a second qubit frequency area, wherein a step is included in which the second qubit frequency and the second qubit frequency area are different from the first qubit frequency and the first qubit frequency area, respectively, and the second qubit diagonally presents to the first qubit in a two-dimensional grid of qubits.SELECTED DRAWING: Figure 1

Description

[Technical Field]

[0001] The present disclosure relates to frequency patterns for reducing parasitic interactions in qubit grids. [Background technology]

[0002] Large-scale quantum computers have the potential to provide fast solutions to several classes of difficult problems. For large-scale quantum computing to be feasible, several challenges must be overcome in the design and implementation of quantum architectures for controlling and programming quantum hardware. Reducing the complexity of quantum architectures while maintaining a high level of control over the qubits contained therein is a crucial step in building scalable quantum computers. [Prior art documents] [Non-patent literature]

[0003] [Non-Patent Document 1] "Fast adiabatic qubit gates using only σZ control", J. Martinis and M. Geller, Phys. Rev. A 90, 022307 (2014) [Non-patent document 2] "Surface codes: Towards practical large-scale quantum computation", A. Fowler et al., Phys. Rev. A 86, 032324 (2012) Summary of the Invention [Means for solving the problem]

[0004] This specification relates to reducing parasitic interactions between qubits. For example, this specification describes systems and methods for operating qubits through frequency patterns that reduce parasitic interactions, for example, during idling, measurement, and application of quantum logic gates.

[0005] In general, one inventive aspect of the subject matter described herein may be embodied in a method for operating a system of qubits, the method including the act of operating the system of qubits, the system of qubits including a first plurality of qubits, each qubit in the first plurality configured to operate at a qubit frequency derived from one of a plurality of first qubit frequency ranges, a second plurality of qubits, each qubit in the second plurality configured to operate at a qubit frequency derived from one of a plurality of second qubit frequency ranges, and a plurality of qubit couplers. and a plurality of qubit couplers, each qubit coupler defining a nearest neighbor interaction between a corresponding qubit from a first plurality of qubits and a corresponding qubit from a second plurality of qubits, the system of qubits arranged as a two-dimensional grid, each qubit from the first plurality of qubits coupled to a plurality of qubits from the second plurality of qubits via the plurality of qubit couplers; and operating the system of qubits includes operating a first qubit from the first plurality of qubits at a first qubit frequency from a first qubit frequency range and operating a second qubit from the first plurality of qubits at a second qubit frequency from a second qubit frequency range, the second qubit frequency and the second qubit frequency range being different from the first qubit frequency and the first qubit frequency range, respectively, and the second qubit being diagonal to the first qubit in the two-dimensional grid.

[0006] Other implementations of this aspect include corresponding computer systems, devices, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the method. One or more computer systems may be configured to perform particular operations or actions by having installed thereon software, firmware, hardware, or a combination thereof that causes the system to perform the actions during operation. One or more computer programs may be configured to perform particular operations or actions by including instructions that, when executed by a data processing device, cause the device to perform the actions.

[0007] These and other implementations may each optionally include one or more of the following features, alone or in combination: In some implementations, the plurality of first qubits comprises data qubits and the plurality of second qubits comprises measurement qubits.

[0008] In some implementations, the method further includes operating a third data qubit from the plurality of data qubits at a second data qubit frequency, the third data qubit being different from the second data qubit and diagonal to the first data qubit in the two-dimensional grid.

[0009] In some implementations, the method further includes operating a fourth data qubit from the plurality of data qubits at a fourth data qubit frequency derived from a third data qubit frequency range, where the third data qubit frequency range is different from the first data qubit frequency range and where the fourth data qubit is different from the second and third data qubits and is diagonal to the first data qubit in the two-dimensional grid; and operating a fifth data qubit from the plurality of data qubits at a fifth data qubit frequency derived from the third data qubit frequency range, where the fifth data qubit is different from the second, third, and fourth data qubits and is diagonal to the first data qubit in the two-dimensional grid.

[0010] In some implementations, the third data qubit frequency domain is the same as the second data qubit frequency domain.

[0011] In some implementations, the third data qubit frequency domain is different from the second data qubit frequency domain.

[0012] In some implementations, the first data qubit frequency and the second data qubit frequency differ by 2η, where η represents the system nonlinearity.

[0013] In some implementations, the first data qubit frequency differs from either (i) the third data qubit frequency, (ii) the fourth data qubit frequency, or (iii) the fifth data qubit frequency by 2η, where η represents the system nonlinearity.

[0014] In some implementations, η=0.2 GHz.

[0015] In some implementations, the second data qubit frequency range comprises a predetermined frequency range, possibly including a frequency range that is 10 MHz wide.

[0016] In some implementations, the difference between the second data qubit frequency and the third data qubit frequency in the second frequency range is greater than a next nearest neighbor coupling strength g of the coupling between the second data qubit and the third data qubit.

[0017] In some implementations, the third data qubit frequency range comprises a predetermined frequency range, possibly including a frequency range that is 10 MHz wide.

[0018] In some implementations, the difference between the fourth data qubit frequency and the fifth data qubit frequency in the third frequency region is greater than the next-nearest neighbor coupling strength g of the coupling between the fourth data qubit and the fifth data qubit.

[0019] In some implementations, the plurality of data qubit frequency ranges comprises four data qubit frequency ranges, the four data qubit frequency ranges possibly including a first idling frequency range, a first echo operation frequency range, a first single-qubit gate frequency range, and a first interaction frequency range.

[0020] In some implementations, operating the system of qubits further includes operating a first measurement qubit from the plurality of measurement qubits at a first measurement qubit frequency derived from a first measurement qubit frequency range, and operating a second measurement qubit from the plurality of measurement qubits at a second measurement qubit frequency derived from a second measurement qubit frequency range, wherein the second measurement qubit frequency and the second measurement qubit frequency range are different from the first measurement qubit frequency and the first measurement qubit frequency range, respectively, and the second measurement qubit is diagonal to the first measurement qubit in the two-dimensional grid.

[0021] In some implementations, the method further includes operating a third measurement qubit from the plurality of measurement qubits at a second measurement qubit frequency, the third measurement qubit being different from the second measurement qubit and diagonal to the first measurement qubit in the two-dimensional grid.

[0022] In some implementations, the method further includes operating a fourth measurement qubit from the plurality of measurement qubits at a fourth measurement qubit frequency derived from a third measurement qubit frequency range, where the third measurement qubit frequency range is different from the first measurement qubit frequency range and the fourth measurement qubit is different from the second measurement qubit and the third measurement qubit and is diagonal to the first measurement qubit in the two-dimensional grid; and operating a fifth measurement qubit from the plurality of measurement qubits at a fifth measurement qubit frequency derived from the third measurement qubit frequency range, where the fifth measurement qubit is different from the second measurement qubit, the third measurement qubit, and the fourth measurement qubit and is diagonal to the first measurement qubit in the two-dimensional grid.

[0023] In some implementations, the third measurement qubit frequency range is the same as the second measurement qubit frequency range.

[0024] In some implementations, the third measurement qubit frequency range is different from the second measurement qubit frequency range.

[0025] In some implementations, the first measurement qubit frequency and the second measurement qubit frequency differ by 2η, where η represents the system nonlinearity.

[0026] In some implementations, the first measurement qubit frequency differs from either (i) the third measurement qubit frequency, (ii) the fourth measurement qubit frequency, or (iii) the fifth measurement qubit frequency by 2η, where η represents the system nonlinearity.

[0027] In some implementations, η=0.2 GHz.

[0028] In some implementations, the second measurement qubit frequency range comprises a predetermined frequency range, possibly including a frequency range that is 10 MHz wide.

[0029] In some implementations, the difference between the second measurement qubit frequency and the third measurement qubit frequency in the second frequency range is greater than the next-nearest neighbor coupling strength g of the coupling between the second measurement qubit and the third measurement qubit.

[0030] In some implementations, the third measurement qubit frequency range comprises a predetermined frequency range, possibly including a frequency range that is 10 MHz wide.

[0031] In some implementations, the difference between the fourth measurement qubit frequency and the fifth measurement qubit frequency in the third frequency region is greater than the next-nearest neighbor coupling strength g of the coupling between the fourth measurement qubit and the fifth measurement qubit.

[0032] In some implementations, the plurality of measurement qubit frequency ranges comprises four measurement qubit frequency ranges, and the four measurement qubit frequency ranges optionally include a first idling frequency range, a first echo operation frequency range, a first single-qubit gate frequency range, and a first interaction frequency range.

[0033] In some implementations, the plurality of measurement qubit frequency ranges and the plurality of data qubit frequency ranges further comprise a readout and reset frequency range adjacent to one of the plurality of measurement qubit frequency ranges.

[0034] In some implementations, an apparatus for operating a system of qubits includes a first plurality of qubits, wherein each qubit in the first plurality is configured to operate at a qubit frequency derived from one of a plurality of first qubit frequency ranges; a second plurality of qubits, wherein each qubit in the second plurality is configured to operate at a qubit frequency derived from one of a plurality of second qubit frequency ranges; and a plurality of qubit couplers, wherein each qubit coupler in the plurality of qubits defines a nearest neighbor interaction between a corresponding qubit from the first plurality and a corresponding qubit from the second plurality of qubits, wherein the system of qubits is arranged as a two-dimensional grid, and The system includes a plurality of qubit couplers, each qubit coupled to a plurality of qubits of a second plurality of qubits via a plurality of qubit couplers, and a qubit controller module configured to operate the system of qubits, wherein operating the system of qubits includes operating a first qubit from the first plurality of qubits at a first qubit frequency derived from a first qubit frequency range, and operating a second qubit from the first plurality of qubits at a second qubit frequency derived from a second qubit frequency range, wherein the second qubit frequency and the second qubit frequency range are different from the first qubit frequency and the first qubit frequency range, respectively, and the second qubit is diagonal to the first qubit in a two-dimensional grid.

[0035] In some implementations, the qubit controller module comprises an excitation pulse generator and one or more excitation drive lines, and operating the qubit at a qubit frequency derived from the qubit frequency domain comprises controlling the qubit via excitation pulses on the excitation drive lines.

[0036] In some implementations, the one or more excitation drive lines comprise a global excitation drive line.

[0037] In some implementations, the first plurality of qubits comprises data qubits and the second plurality of qubits comprises measurement qubits.

[0038] The subject matter described herein can be implemented in particular embodiments to realize one or more of the following advantages.

[0039] Quantum computing systems that implement methods for reducing parasitic interactions between qubits, as described herein, can perform quantum computing operations while reducing parasitic interactions between qubits and introducing minimal errors. The methods described herein can improve the robustness of quantum computing systems and improve the accuracy of computations performed by the quantum computing systems.

[0040] The methods for reducing parasitic interactions between qubits as described herein are scalable, allowing for generous and practical requirements on the physical quantum computing hardware needed to implement the methods and perform quantum computations. For example, the methods and systems described herein may be implemented using a qubit frequency control architecture.

[0041] Furthermore, methods for reducing parasitic interactions between qubits as described herein can increase the efficiency of computations performed by quantum computing systems implementing the methods. For example, in some cases, the methods may allow several quantum logic gates to be implemented simultaneously, thus reducing the time required to execute an algorithm.

[0042] Additionally, methods for reducing parasitic interactions between qubits in quantum computing systems as described herein may be enhanced by placing two echo pulses (rotations about the X-axis and / or Y-axis, designed to reduce the qubit's sensitivity to the environment) on an idle qubit between entangling operations on two other qubits, greatly simplifying the algorithmic implementation of quantum computations performed by the quantum computing system. For example, two echo pulses may be used to suppress noise and construct sequences with ideal identity unitary. Without the ability to place two echo pulses on an idle qubit between entangling operations on two other qubits, it may be necessary to modify the algorithmic implementation to account for the unitary nature of the echo pulses, which are interchangeable through an entanglement gate representing the entanglement operation. The systems and methods described herein may avoid this modification.

[0043] Furthermore, the methods described herein for reducing parasitic interactions between qubits are tolerant to the required qubit detuning and threshold for parasitic coupling strength, thus giving the methods described herein widespread utility and applicability.

[0044] One approach for building and operating quantum computing devices is based on surface codes operated as stabilizer codes. Surface codes provide a practical way to identify and handle errors in two-dimensional arrays of qubits. However, standard implementations of surface codes, such as those described herein, require nearest-neighbor entangling operations in dense patterns. Such dense patterns can cause parasitic coupling between qubits diagonally opposite each other, as described herein.

[0045] A quantum computing system implementing a method for reducing parasitic interactions between qubits as described herein may perform surface code cycles using a particular configuration of paired qubits. This configuration may enable the surface code to be reliably implemented using a dense, two-dimensional grid of closely spaced qubits. Additionally, this configuration may enable the surface code to be implemented using fewer layers of entanglement operations compared to other surface code implementations.

[0046] The details of one or more implementations of the subject matter herein are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter will become apparent from the description, drawings, and claims. [Brief explanation of the drawings]

[0047] [Figure 1] FIG. 1 is a block diagram of an exemplary quantum computing system. [Figure 2] FIG. 1 is a flow diagram of an exemplary process for operating a system of qubits. [Figure 3A] FIG. 1 illustrates an exemplary schematic data qubit frequency pattern. [Figure 3B]1 is an exemplary plot showing idling error due to parasitic interactions versus qubit frequency divided by system nonlinearity for a single qubit. [Figure 4] FIG. 1 illustrates exemplary data qubit and measurement qubit frequencies. [Figure 5] FIG. 1 is a flow diagram of an exemplary process for performing an entanglement operation on a two-dimensional array of qubits. [Figure 6] FIG. 1 illustrates an exemplary pairing of data and measurement qubits to perform entanglement operations on a two-dimensional array of qubits. [Figure 7] 10 is an exemplary plot of a controlled-Z quantum gate frequency trajectory. [Figure 8] 1 is an exemplary plot of the probability of parasitic occupation transfer versus diagonal coupling strength. [Figure 9] FIG. 10 illustrates an exemplary quantum circuit for measuring stabilizers during a surface code detection cycle. [Figure 10] FIG. 10 is a flow diagram of an exemplary process for performing a surface signature error detection cycle. [Figure 11] FIG. 10 illustrates an exemplary uniform stabilizer for a surface code detection cycle. DETAILED DESCRIPTION OF THE INVENTION

[0048] Like reference numbers and designations in the various drawings indicate like elements.

[0049] Qubits in quantum computing systems can be arranged and operated in a two-dimensional grid using nearest-neighbor interactions. However, in such a grid, coupling between qubits diagonally opposite each other is relatively large due to their proximity. Such coupling is parasitic and unintended and uncontrolled. For example, if two diagonal qubits form a parasitic coupling, the qubits may affect each other in an unintended and uncontrolled manner, for example, by inducing unwanted transitions in one or both of the qubits. An unwanted transition in a qubit can flip the state of the qubit, for example, from one computation state to another, or cause a transition to a higher qubit level outside the computation subspace. Such a transition can introduce errors into the computation performed by the qubit. Therefore, minimizing parasitic coupling between qubits is a key challenge in quantum computing when operating quantum computers, especially at large scales.

[0050] Example Operating Environment 1 is a block diagram of an exemplary quantum computing system 100. The exemplary quantum computing system 100 includes a plurality of qubits 102 (represented as open and filled circles) and a qubit controller module 104. The exemplary quantum computing system 100 is an example of a system that may be used to perform quantum algorithmic operations, simulations, or calculations.

[0051] The multiple qubits 102 are arranged in a two-dimensional grid 112. For clarity, the two-dimensional grid 112 shown in FIG. 1 includes 7×7 qubits, although in some implementations, the system 100 may include fewer or more qubits. The multiple qubits 102 interact with one another via multiple qubit couplers, e.g., qubit coupler 114. The multiple qubit couplers define nearest-neighbor interactions between the multiple qubits 102. In some implementations, the strength g of the multiple qubit couplers is an adjustable parameter. In some cases, the multiple qubit couplers included in the quantum computing system 100 may be couplers with fixed coupling strengths.

[0052] In some implementations, the plurality of qubits 102 may include data qubits, e.g., open circles, such as qubit 118, and measurement qubits, e.g., filled circles, such as qubit 116. A data qubit is a qubit that participates in a computation being performed by system 100. A measurement qubit is a qubit that can be used to determine the result of a computation performed by a data qubit. That is, during the computation, the unknown state of the data qubit is transferred, using a suitable physical operation, to a measurement qubit and measured via a suitable measurement operation performed on the measurement qubit.

[0053] The plurality of data qubits are each configured to operate at a qubit frequency derived from a respective frequency range. For example, each data qubit may be configured to operate at a respective data qubit frequency derived from one of the plurality of data qubit frequency ranges. Similarly, each measurement qubit may be configured to operate at a respective measurement qubit frequency derived from one of the plurality of measurement qubit frequency ranges.

[0054] A qubit frequency domain may include a computational qubit frequency domain. For example, each data or measurement qubit may be configured to operate at a qubit frequency derived from a respective computational qubit frequency domain, e.g., when the qubit is involved in a computational or algorithmic operation. If transitions between qubit computational basis states occur at the qubit frequency, the qubit may be said to operate at a qubit frequency derived from the computational qubit frequency domain.

[0055] The qubit frequency range may include an idle qubit frequency range. For example, each data or measurement qubit may be configured to operate at a qubit frequency derived from its respective idle qubit frequency range, e.g., when the qubit is idle and not involved in an interaction or computational operation. If the intended unitary for the qubit is unity II for a set duration, the qubit may be said to operate at a qubit frequency derived from the idle qubit frequency range for that duration. Thus, the qubit is not involved in an entangled quantum logic gate operation, and only echo pulses with unity unitary are applied to the qubit, in which case the echo pulses are defined rotations about the X and / or Y axes designed to reduce the qubit's sensitivity to the environment. In other words, while other qubits perform interactions or undergo unitary operations, the qubit may be said to be dormant at a frequency derived from the idle qubit frequency range that minimizes interactions between the qubit and other qubits.

[0056] The qubit frequency range may include an interaction qubit frequency range, e.g., the data and measurement qubits may be configured to operate at respective qubit frequencies derived from the respective interaction frequency ranges, e.g., when the data and measurement qubits interact.

[0057] Qubit frequency ranges may include readout and reset frequency ranges. For example, a measurement qubit may be configured to operate at a qubit frequency derived from the respective readout and reset frequency ranges, e.g., when a measurement operation is being performed on the measurement qubit. A qubit may be said to operate at a qubit frequency derived from the readout and reset qubit frequency ranges when the qubit frequency is close to or matches the operating frequency of the readout resonator or other measurement device to effect the measurement or reset operation.

[0058] The data qubits and measurement qubits are arranged such that the qubit couplers define nearest neighbor interactions between the data qubits and the measurement qubits, i.e., each data qubit is coupled to multiple measurement qubits and each measurement qubit is coupled to multiple data qubits.

[0059] In other implementations, the plurality of qubits 102 may not be separated into a plurality of data qubits and a plurality of measurement qubits. For example, if system 100 includes a quantum computer that does not implement a surface code, e.g., a quantum computer used to execute one or more quantum algorithms, such as the Supremacy algorithm, the system may not need to distinguish between data qubits and measurement qubits. In these examples, the disclosed systems and methods may be used to reduce the number of layers of quantum logic gates required to execute a quantum algorithm, thus increasing the speed of the algorithm and reducing errors in the algorithm.

[0060] The same process described herein may be used when there is no distinction between data qubits and measurement qubits. For example, the qubits may be configured to operate at qubit frequencies within respective frequency ranges. Some qubits may be configured to operate at respective qubit frequencies within one of a plurality of different first qubit frequency ranges. Other qubits may be configured to operate at respective qubit frequencies within one of a plurality of different second qubit frequency ranges. The qubit frequency ranges in the plurality of first and second qubit frequency ranges may include respective calculation frequency ranges, idle frequency ranges, interaction frequency ranges, and readout / reset frequency ranges.

[0061] As described herein, the exemplary two-dimensional grid 112 may include parasitic couplings between qubits diagonally opposite one another, e.g., parasitic couplings 120. In some cases, the parasitic couplings between qubits may have a non-zero parasitic coupling strength g diag For example, in some cases, the parasitic coupling strength is g diag It can take values ​​from / π to 5MHz.

[0062] The plurality of qubits 102 in the two-dimensional grid 112 are operated via a qubit controller module 104. The qubit controller module 104 may operate the qubits 102, for example, by controlling the frequency of the qubits 102 according to a qubit operating frequency 108. The qubit operating frequency depends on the type of qubits included in the system 100 and the operation being performed by the system. Exemplary qubit operating frequency patterns for reducing parasitic interactions between diagonal qubits are described in detail below with reference to FIGS. 2-4.

[0063] For example, the qubit controller module 104 may control the individual frequencies of the qubits 102 such that the frequency of one or more of the qubits is adjusted toward or away from the frequency of excitation pulses generated by the excitation pulse generator 110 on the excitation drive line 124, e.g., the entanglement operation frequency 106. The excitation pulses generated by the excitation pulse generator 110 may include pulses whose frequencies implement quantum operations, e.g., quantum logic gates. For example, the excitation pulse generator 110 may be configured to generate excitation pulses whose frequencies cause one or more qubits to perform an entanglement operation, e.g., a controlled Z-gate. Performing entanglement operations on a two-dimensional grid of qubits is described in more detail below with reference to FIGS. 2-5.

[0064] Multiple qubits may be coupled to excitation drive lines via respective couplers, e.g., coupler 126. In some cases, the couplers may be capacitive couplers, e.g., realized by microwave lines running adjacent to the qubit capacitors. For convenience, a global excitation drive line is shown in FIG. 1. However, in some implementations, system 100 may include multiple excitation drive lines, e.g., corresponding to multiple qubits.

[0065] Qubit controller module 104 may be configured to adjust the frequency of qubit 102 via one or more qubit frequency control lines, e.g., qubit frequency control line 122. For convenience, one qubit frequency control line is shown in FIG. 1 . However, in some implementations, system 100 may include multiple qubit frequency control lines, e.g., corresponding to each of multiple qubits 102. The qubit frequency control lines may be supplied by in-plane wiring or out-of-plane wiring.

[0066] The type of qubit controller 104 utilized by system 100 depends on the type of qubits the system uses. As an example, qubits realized via atomic, molecular, or solid-state quantum systems typically have associated qubit-level energy separations in the microwave or optical domain. The states of such qubits can be manipulated and controlled using external fields, such as microwave or optical fields. In such cases, as an example, mode-locked lasers can serve as qubit controllers due to their broadband optical spectrum that characterizes both radio frequency and microwave structures. In another example, the qubit controller can include a collection of individual qubit controllers realized by radio frequency generators, as well as one or a collection of global excitation controllers realized by radio frequency or microwave generators. In either case, the qubit controller can be operated manually or can be connected to a computer and controlled via suitable software that allows the required qubit operations to be specified and executed automatically.

[0067] Programming the Hardware: Qubit Frequency Patterns to Reduce Parasitic Interactions For convenience, the disclosure provided below with reference to Figures 2-4 will be described with reference to a multiple-qubit system including multiple data qubits and multiple measurement qubits that interact via qubit couplers that define nearest-neighbor interactions between the data qubits and measurement qubits. However, this is one example of a qubit system that may be programmed and operated using the techniques described herein. For example, in some implementations, the following techniques may be used to program and operate a qubit system that does not distinguish between data or measurement qubits. For example, if the qubit system is used to perform a quantum algorithm, e.g., a supremacy algorithm, the qubits may be indistinguishable. In settings where distinguishing between data qubits and measurement qubits is not required, the following arrangements and processes may still be used.

[0068] 2 is a flow diagram of an exemplary process 200 for operating a system of qubits. For convenience, process 200 is described as being performed by one or more quantum or classical computer systems located at one or more locations. For example, process 200 may be implemented using qubit controller 104 of system 100 described above with reference to FIG. 1.

[0069] The system of qubits includes qubits that interact via qubit couplers that define nearest-neighbor interactions. In some implementations, the system of qubits may include a plurality of data qubits and a plurality of measurement qubits that interact via qubit couplers that define nearest-neighbor interactions between the data qubits and the measurement qubits. The system of qubits is arranged as a two-dimensional grid, and each data qubit of the plurality of data qubits is coupled to a plurality of measurement qubits via a plurality of qubit couplers. Each data qubit is configured to operate at a data qubit frequency derived from one of a plurality of data qubit frequency ranges. Each measurement qubit is configured to operate at a measurement qubit frequency derived from one of a plurality of measurement qubit frequency ranges. Exemplary data qubit frequency ranges and measurement qubit frequency ranges are described below with reference to steps 202 and 204. An exemplary system of qubits is shown with reference to FIG. 1.

[0070] The system operates a first data qubit from the plurality of data qubits at a first data qubit frequency from a first data qubit frequency range (step 202). For example, as shown in schematic data qubit frequency pattern 300 described below with reference to Figure 3A, data qubit 302 may be operated at data qubit frequency b from a respective data qubit frequency range B.

[0071] The system operates a second data qubit from the plurality of data qubits at a second data qubit frequency from a second data qubit frequency range (step 204). The second data qubit is a data qubit diagonal to the first data qubit in the two-dimensional grid. For example, as shown in schematic data qubit frequency pattern 300 described below with reference to FIG. 3A, the system may operate a second qubit, e.g., qubit 304, at a data qubit frequency a from a respective data qubit frequency range A.

[0072] The second data qubit frequency and the second data qubit frequency range are different from the first data qubit frequency and the first data qubit frequency range, respectively. For example, as shown in schematic data qubit frequency pattern 300 described below with reference to Figure 3A, a first data qubit 302 may be operated at a data qubit frequency b from a respective data qubit frequency range B, and a second data qubit 304 diagonally opposite to data qubit 302 may be operated at a data qubit frequency a from a different data qubit frequency range A.

[0073] In some implementations, the system may further operate a third data qubit from the plurality of data qubits at a second data qubit frequency. The third data qubit is different from the second data qubit and is diagonal in two dimensions to the first data qubit. For example, as shown in schematic data qubit frequency pattern 300 described below with reference to FIG. 3A , the system may operate a first data qubit 302 at a data qubit frequency b from a respective data qubit frequency region B, operate a second data qubit 304 that is diagonal to data qubit 302 at a data qubit frequency a from a different data qubit frequency region A, and operate a third data qubit 306 that is different from the second data qubit 304 and is diagonal to the first data qubit 302.

[0074] In some implementations, the system may further operate a fourth data qubit and a fifth data qubit from the plurality of data qubits at respective fourth and fifth data qubit frequencies derived from a third data qubit frequency range, the third data qubit frequency range being different from the first data qubit frequency range, and the fourth and fifth data qubits being diagonal to the first data qubit.

[0075] 3A , the system may operate a first data qubit 302 at a data qubit frequency b from a respective data qubit frequency range B, a second data qubit 304 at a data qubit frequency a from a data qubit frequency range A, a third data qubit 306 at a data qubit frequency a′ from a data qubit frequency range A, a fourth data qubit 310 at a data qubit frequency a from a data qubit frequency range A, and a fifth data qubit 308 at a data qubit frequency a′ from a data qubit frequency range A. In some implementations, the third data qubit frequency range may be the same as the second data qubit frequency range. In other implementations, the third data qubit frequency range may be different from the second data qubit frequency range, e.g., data qubits 308 and 310 may operate at data qubit frequencies c′ and c, respectively.

[0076] As shown in the schematic data qubit frequency pattern 300 of FIG. 3A , in some implementations, the multiple data qubit frequency regions include two data qubit frequency regions, e.g., a first region A and a second region B. In these implementations, the data qubit frequencies from each region may be offset by −2η, where η represents the nonlinearity of the system. For example, data qubit frequencies a∈A and b∈B may differ by 2η. One explanation for why the frequency difference depends on the nonlinearity of the system is as follows: Idle qubits may be affected by parasitic interactions from diagonal qubits. To minimize these effects, the frequency difference between diagonal qubits is optimally selected. The effects of diagonal qubits may be represented by the nonlinearity of the system. Therefore, by analyzing the nonlinearity of the system, a range of various frequency values ​​that locally minimizes the effects of diagonal qubits can be discovered. That is, the system nonlinearity provides instructions on how to minimize parasitic interactions between qubits and, therefore, how to minimize errors in the system. An example plot showing the idling ZZ error from frequency shifts due to parasitic interactions from the diagonal qubit (y-axis) versus the qubit frequency divided by the system nonlinearity (x-axis) for a single qubit is shown in Figure 3B.

[0077] Additionally, in some implementations, data qubit frequencies from a particular data qubit frequency range may include frequencies within a predetermined frequency range, e.g., a predetermined frequency range that is 10 MHz wide. For example, data qubits 304 and 306 in Figure 3A may operate at data qubit frequencies a and a', respectively, where a and a' differ by about 10 MHz.

[0078] Furthermore, in some implementations, swapping may be avoided by ensuring that the difference between data qubit frequencies within a predetermined frequency range is greater than the next-nearest-neighbor coupling constant g. For example, the difference between the data qubit frequencies a, a' of qubits 304 and 306, which are diagonal to qubit 302, may be greater than g, e.g., (a-a')>>g. With a next-nearest-neighbor coupling constant g=1 MHz, a detuning between a and a' of 10 MHz is tolerable.

[0079] As shown in schematic data qubit frequency pattern 350 of Figure 3A, in some implementations, the multiple data qubit frequency regions include four data qubit frequency regions, e.g., regions A, B, C, and D. The multiple frequency regions allow qubits to be "parked" and individually controlled, e.g., using global XY excitation drive lines. For example, at η = 200 MHz, data qubits can be parked or operated at frequencies between 6 GHz and 7 GHz, e.g., at 6.7 GHz, 6.3 GHz, 6.8 GHz, and 6.2 GHz.

[0080] In these implementations, the plurality of data qubit frequency ranges may include a first idle frequency range, and when the data qubit is not actively participating in an algorithmic computation being performed by the system of qubits and is idling, the data qubit may be configured to operate at the idle frequency.

[0081] The plurality of data qubit frequency ranges may further include a first echo operation frequency range. The data qubit may be configured to operate at an echo operation frequency when an echo operation is being performed on the data qubit.

[0082] The plurality of data frequency regions may further include a first single-qubit gate frequency region. When a single-qubit quantum gate, for example a Hadamard quantum logic gate or a Pauli X, Y, or Z quantum logic gate, is performed on a data qubit, the data qubit may be configured to operate in a single-qubit gate frequency region.

[0083] The plurality of data frequency regions may further include an interaction frequency region. When a data qubit is interacting with a neighboring measurement qubit, for example, when an entanglement operation is performed on the paired data and neighboring measurement qubits, the data qubit may be configured to operate in the interaction frequency region. Exemplary data qubit frequency regions are shown with reference to FIG. 4.

[0084] In implementations in which the plurality of data qubit frequency ranges includes four data qubit frequency ranges, operating the system of qubits may include, for each data qubit, operating the data qubit at a data qubit frequency derived from the data qubit frequency range, where each other data qubit diagonally opposite the data qubit is operated at a respective other data qubit frequency derived from a different data qubit frequency range, and where each other data qubit diagonally opposite the data qubit is operated at a respective other data qubit frequency derived from a different data qubit frequency range.

[0085] 3A , data qubit 352 may be operated at a data qubit frequency b from respective data qubit frequency region B, and each other data qubit diagonally opposite data qubit 352, e.g., data qubits 354, 356, 358, and 360, may be operated at data qubit frequencies a, a′, c, or c′ from different data qubit frequency regions A and C. Other data qubits diagonally opposite data qubit 352, e.g., data qubits 356 and 358, or data qubits 354 and 360, operate at respective data qubit frequencies from different data qubit frequency regions. That is, data qubit 356 operates at a frequency from data qubit frequency region A, and data qubit 358 operates at a frequency from different data qubit frequency region C. Similarly, data qubit 354 operates at a frequency derived from data qubit frequency range A, and data qubit 360 operates at a frequency derived from a different data qubit frequency range C.

[0086] In these implementations, the data qubit frequencies corresponding to the diagonal data qubits may be offset by −2η, e.g., (ab) / η≧2. For example, in schematic data qubit frequency pattern 350, the data qubit frequencies may be set as a=0, b=−2η, c=0.5η, and d=−2.5η. In some implementations, η=0.2 GHz, providing a frequency range of approximately 0.8-1.0 GHz in the frequency domain in which similar qubits, data, or measurement qubits, may be parked. For example, if all data qubits are parked at frequencies between 6-7 GHz, e.g., in the range of 0.8-1 GHz, and all measurement qubits are parked between 4-5 GHz, interactions may occur between 5 GHz and 6 GHz. However, other layouts are possible.

[0087] Additionally, in some implementations, data qubit frequencies from a particular data qubit frequency range may include frequencies within a predetermined frequency range, e.g., a predetermined frequency range that is 10 MHz wide. For example, data qubits 354 and 356 in Figure 3A may operate at data qubit frequencies a and a', respectively, where a and a' differ by about 10 MHz.

[0088] Furthermore, in some implementations, the difference between other data qubit frequencies diagonal to the qubit, e.g., data qubit frequencies a, a', or c, c', may be greater than the nearest neighbor coupling constant g, e.g., (a-a')>>g.

[0089] For example, as described herein with reference to Figure 4, the data qubit frequency patterns described above facilitate entanglement operations, e.g., dense patterns in controlled Z quantum logic gates, to be performed without nearest-neighbor qubits (including diagonally) at the same frequency. Thus, parasitic interactions may be reduced.

[0090] The properties of the data qubits described above may also apply to multiple measurement qubits in a system of qubits. For example, operating the system of qubits may further include, for each measurement qubit, operating the measurement qubit at a measurement qubit frequency derived from a measurement qubit frequency range, where each other measurement qubit diagonally opposite the measurement qubit is operated at a respective other measurement qubit frequency derived from a different measurement qubit frequency range.

[0091] In some implementations, the multiple measurement qubit frequency ranges include two measurement qubit frequency ranges. The measurement qubit frequency and the other measurement qubit frequency may differ by 2η. The other measurement qubit frequency may include a frequency within a predetermined frequency range, including a predetermined frequency range that is 10 MHz wide in some cases. The difference between the other measurement qubit frequencies within the predetermined frequency range may be greater than the nearest neighbor coupling constant g.

[0092] In some implementations, the multiple measurement qubit frequency ranges may include four measurement qubit frequency ranges, possibly including a second idle frequency range, a second echo operation frequency range, a second single-qubit gate frequency range, and a second interaction frequency range. For example, the data qubit may be parked or operated at a frequency between 6 GHz and 7 GHz, the measurement qubit may be parked or operated at a frequency between 4 GHz and 5 GHz, and the interaction between the qubits may occur between 5 GHz and 6 GHz.

[0093] In these examples, operating the system of qubits may further include, for each measurement qubit, operating the measurement qubit at a measurement qubit frequency derived from a measurement qubit frequency region, where each other measurement qubit diagonally opposite the measurement qubit is operated at a respective other measurement qubit frequency derived from a different measurement qubit frequency region, and where each other measurement qubit diagonally opposite the measurement qubit is operated at a respective other measurement qubit frequency derived from a different measurement qubit frequency region.

[0094] In some implementations, diagonal measurement qubits may be offset by 2η, and in some cases η=0.2 GHz. Other measurement qubit frequencies from the same measurement qubit frequency range may include frequencies within predetermined frequency ranges, including predetermined frequency ranges that are 10 MHz wide. The difference between other measurement qubit frequencies from the same measurement qubit frequency range is greater than g.

[0095] In some implementations, a readout and reset frequency range may be included adjacent to one of the multiple measurement qubit frequency ranges. By locating the readout and reset frequency range adjacent to the measurement qubit frequency range, the frequency of the qubit can be tuned close to the readout resonator to obtain a large dispersion and therefore a large measurement signal. In addition, by locating the readout and reset frequency range adjacent to the measurement qubit frequency range, a reset operation of the measurement qubit using the readout resonator is possible. Furthermore, by locating the readout and reset frequency range adjacent to the measurement qubit frequency range, movement of the measurement qubit beyond the readout resonator is possible, allowing movement of the data qubit close to the readout resonator to obtain a large dispersion and a large signal without adversely affecting the measurement qubit.

[0096] 3A shows exemplary schematic data qubit frequency patterns 300 and 350. Exemplary schematic data qubit frequency pattern 300 shows a plurality of data qubits, e.g., data qubits 302, 304, 306, 308, and 310, coupled to a plurality of measurement qubits via nearest-neighbor interactions. Exemplary schematic data qubit frequency pattern 300 shows data qubits operated at data qubit frequencies a, a′, b, and b′ from two data qubit frequency regions A and B. Each data qubit in exemplary schematic data qubit frequency pattern 300 is operated at a data qubit frequency that is different from the data qubit frequency at which its diagonally adjacent data qubit operates. For example, data qubit 302 operates at data qubit frequency b from frequency region B, while its diagonally adjacent data qubits 304, 306, 308, and 310 operate at data qubit frequencies a, a′, a′, and a, respectively, from frequency region A.

[0097] Exemplary schematic data qubit frequency pattern 350 shows a plurality of data qubits, e.g., data qubits 352, 354, 356, 358, and 360, coupled to a plurality of measurement qubits via nearest-neighbor interactions. Exemplary schematic data qubit frequency pattern 350 shows data qubits operated at data qubit frequencies a, a', b, b', c, c', and d, d'. The data qubit frequencies may be frequencies from four respective data qubit frequency regions, e.g., regions A, B, C, and D. Each data qubit in exemplary schematic data qubit frequency pattern 350 is operated at a data qubit frequency that is different from the data qubit frequency at which its diagonally adjacent data qubit operates. In addition, qubits diagonally adjacent to a data qubit and diagonally opposite each other, e.g., qubits 356 and 358, operate at different data qubit frequencies. For example, data qubit 352 operates at data qubit frequency b from frequency region B, while its diagonally adjacent data qubits 354 and 360 are diagonally opposite each other and therefore operate at data qubit frequencies a and c′ from frequency regions A and C, respectively. Similarly, its diagonally adjacent data qubits 356 and 358 are diagonally opposite each other and therefore operate at data qubit frequencies a′ and c from frequency regions A and C, respectively.

[0098] 4 shows exemplary data qubit and measurement qubit frequencies 400. The exemplary data qubit and measurement qubit frequencies include nine different frequencies spanning frequency ranges 410, 412, and 414. Four frequencies are data qubit frequencies 402. Four frequencies are measurement qubit frequencies 406. One of the data qubit frequencies is an interaction frequency 404. Similarly, one of the measurement qubit frequencies is an interaction frequency 404. One frequency is a readout and reset frequency 408. This configuration of frequencies allows entanglement operations, e.g., dense patterns in a controlled-Z quantum logic gate, to be performed without neighboring qubits (including diagonally) at the same frequency. Thus, parasitic interactions may be reduced because the qubits are sufficiently geometrically separated.

[0099] Data qubit frequencies 1 and 2 in frequency range 410 are idle data qubit frequencies. Similarly, measurement qubit frequencies 3 and 4 in frequency range 412 are idle measurement qubit frequencies. Data qubit frequencies 3 and 4 and measurement qubit frequencies 1 and 2 in frequency range 414 are qubit frequencies for qubits being manipulated. In some implementations, one or more frequencies in frequency ranges 410 and 412 or in frequency range 414 may also be selected for globally applied single-qubit gates.

[0100] The exemplary data qubit frequency and measurement qubit frequency 400 further includes an additional readout and reset frequency 416 for the data qubit. The additional readout and reset frequency 416 is located at a higher frequency than the data qubit frequency 402, i.e., with reference to the example 400, is shown as being above the data qubit frequency 402. The additional readout and reset frequency 416 may allow the data qubit to be readout or reset while interacting with another pair of qubits. This may be beneficial in a variety of settings. An exemplary setting includes performing surface code error detection in superconducting hardware, because when an over-excited (leaky) state is removed, the measurement qubit becomes a data qubit, and vice versa. In this setting, measuring the data qubit while interacting with another pair of qubits may be extremely beneficial.

[0101] Programming hardware: Simultaneous qubit detuning to reduce parasitic interactions For convenience, the techniques described with reference to Figures 5-8 relate to a multi-qubit system including multiple data qubits and multiple measurement qubits that interact via qubit couplers that define nearest-neighbor interactions between the data qubits and the measurement qubits. However, this is one example of a qubit system that may be programmed and operated using the following techniques. For example, in some implementations, the following techniques may be used to program and operate a qubit system that does not distinguish between data or measurement qubits. For example, if the qubit system is used to perform a quantum algorithm, such as a supremacy algorithm, the qubits may be indistinguishable. In settings where distinguishing between data qubits and measurement qubits is not required, the following techniques and arrangements may still be used.

[0102] 5 is a flow diagram of an exemplary process 500 for performing entanglement operations using a system of qubits. For convenience, process 500 is described as being performed by one or more quantum or classical computer systems located at one or more locations. For example, process 500 may be implemented using qubit controller module 104 of system 100 described above with reference to FIG. 1. In some implementations, process 500 may be performed in conjunction with the frequency patterns described above with reference to FIGS. 2-4.

[0103] The system of qubits includes a plurality of qubits and a plurality of qubit couplers that define nearest-neighbor interactions among the plurality of qubits. In some implementations, the plurality of qubits may include a plurality of data qubits, a plurality of measurement qubits, and a plurality of qubit couplers that define nearest-neighbor interactions between the data qubits and the measurement qubits. The system of qubits is arranged as a two-dimensional grid, and each data qubit of the plurality of data qubits is coupled to a plurality of measurement qubits via a respective qubit coupler. An exemplary two-dimensional grid is shown above with reference to FIG. 1.

[0104] The system pairs multiple data qubits with respective neighboring measurement qubits (step 502). In some implementations, the system may pair multiple data qubits and measurement qubits into non-overlapping pairs. For example, each data qubit paired with its respective neighboring measurement qubit may not be paired with another neighboring measurement qubit. Similarly, each measurement qubit paired with its respective neighboring data qubit may not be paired with another neighboring data qubit. An exemplary pairing of data qubits and neighboring measurement qubits into non-overlapping pairs is shown in the exemplary two-dimensional qubit grid 600 of FIG. 6.

[0105] Alternatively or additionally, the system may pair multiple data qubits and their respective neighboring measurement qubits into pairs with parallel qubit couplers. For example, with reference to the two-dimensional grid 112 shown above with reference to FIG. 1 , the system may pair data qubits with measurement qubits that are directly above or below the data qubits. In this configuration, the paired data and measurement qubits may be described as having north-south parallel couplers.

[0106] When a system pairs multiple data qubits with their respective neighboring measurement qubits in non-overlapping pairs, the parallel couplers have the same orientation. That is, each measurement qubit can be paired with its respective neighboring data qubit to the north (or south). An exemplary pairing of data qubits and neighboring measurement qubits in non-overlapping pairs with north-south parallel couplers is shown in exemplary two-dimensional grid 600 of FIG. 6. In some cases, for example, when a system pairs multiple data qubits with their respective neighboring measurement qubits in overlapping pairs, the parallel couplers can have different orientations. That is, some measurement qubits can be paired with a first data qubit in the north direction and a second data qubit in the south direction. In exemplary two-dimensional grid 600, qubits are coupled with couplers in only the north direction, as indicated by arrow 626.

[0107] As another example, referring to the two-dimensional grid 112 shown above with reference to Figure 1, the system may pair data qubits with measurement qubits that are immediately to the right or left of the data qubits. In this configuration, the paired data and measurement qubits may be described as having east-west parallel couplers.

[0108] If the system pairs multiple data qubits with their respective neighboring measurement qubits in non-overlapping pairs, the parallel couplers have the same orientation. That is, each measurement qubit may be paired with its respective neighboring data qubit to the west (or east). If the system pairs multiple data qubits with their respective neighboring measurement qubits in overlapping pairs, the parallel couplers may have different orientations. That is, some measurement qubits may be paired with a first data qubit in the west direction and a second data qubit in the east direction. An exemplary pairing of data qubits and neighboring measurement qubits into overlapping pairs with differently oriented east-west parallel couplers is shown in exemplary two-dimensional grid 650 of FIG. 6. In exemplary two-dimensional grid 650, qubits are coupled with couplers in both the east and west directions, as indicated by arrows 628 and 630.

[0109] In some implementations, the system may pair a subset of the plurality of data qubits with their respective neighboring measurement qubits. For example, the system may pair the plurality of data qubits with their respective neighboring measurement qubits such that each paired data qubit and measurement qubit is not adjacent to other paired data qubits and measurement qubits. Exemplary non-adjacent pairs of data qubits and their respective neighboring measurement qubits are shown in the exemplary two-dimensional grid 600 of FIG. 6.

[0110] In some implementations, the system may pair multiple data qubits with their respective neighboring measurement qubits into multiple subsets of paired data and measurement qubits, e.g., the system may repeat the pairing process described above across multiple subsets, e.g., until each qubit in the system of qubits is paired with at least one other qubit.

[0111] In some cases, the paired data and measurement qubits of multiple subsets may include non-overlapping subsets of paired data and measurement qubits. For example, the exemplary two-dimensional grid 600 of FIG. 6 shows multiple non-overlapping subsets of paired data and their respective neighboring measurement qubits. In the exemplary two-dimensional grid 600, each subset includes non-adjacent pairs of paired data and their respective neighboring measurement qubits. In other cases, the paired data and measurement qubits of multiple subsets may include overlapping subsets of paired data and measurement qubits. For example, the exemplary two-dimensional grid 650 of FIG. 6 shows multiple overlapping subsets of paired data and their respective neighboring measurement qubits.

[0112] The system performs an entanglement operation in parallel on each paired data and measurement qubit (step 504). For example, the system may apply a two-qubit quantum logic gate, e.g., a controlled Z quantum logic gate, to each paired data and measurement qubit in parallel. Because variations in the frequency amplitude of the applied entanglement operation may occur, performing an entanglement operation in parallel on each paired data and measurement qubit is understood to mean performing the entanglement operation in parallel on each paired data and measurement qubit to the extent permitted by the hardware used to perform process 500. Exemplary variations are described in more detail below.

[0113] If the system generates multiple subsets of paired data qubits and their respective neighboring measurement qubits, the system may perform an entanglement operation in parallel on each paired data and measurement qubit in each subset, as described above with reference to step 504. To perform an entanglement operation on each data and measurement qubit in a system of qubits, the system may perform the entanglement operation sequentially on the paired data and measurement qubits in each subset. In some implementations, the order in which the system selects subsets on which to perform the entanglement operation may be arbitrary.

[0114] As described above with reference to step 504, due to the configuration of paired qubits, each qubit involved in the entanglement operation (or each qubit involved in the sequential application of one of the entanglement operations on the subset of paired qubits) is either non-adjacent to other qubits involved in the entanglement operation or has the same type on the diagonal. For example, if a data qubit and a neighboring measurement qubit are paired into non-adjacent pairs of data and neighboring measurement qubits, as shown in group 602 of FIG. 6, then the qubits involved in one entanglement operation are not adjacent to qubits involved in other entanglement operations. As another example, if a data qubit and a neighboring measurement qubit are paired into overlapping subsets of paired data and measurement qubits with parallel couplers, as shown in group 608 of FIG. 6, then the qubits involved in one entanglement operation have the same type on the diagonal, e.g., qubits 610 and 612.

[0115] Thus, when entangling operations are performed in parallel on each paired data and measurement qubit, each measurement qubit can be detuned without crossing the resonance of another measurement qubit that is executing a similar frequency trajectory on its corresponding data qubit. In fact, because the entangling operations are performed in parallel, the detuning Δf between diagonal qubits is constant (or nearly constant, see below), and therefore no population shifts from diagonal interactions will occur. In addition, each data qubit can execute a portion of a trajectory and need not remain at a constant frequency. For example, the data qubit may execute a frequency trajectory that shifts toward the measurement qubit. The advantages of the method performed by the system remain the same.

[0116] To perform the entanglement operation of each paired data and measurement qubit in parallel, the system detunes each measurement qubit in the paired data and measurement qubit in parallel. As described herein, detuning each measurement qubit in the paired data and measurement qubit in parallel may include maintaining a constant or nearly constant detuning Δf between the measurement qubits in the paired data and measurement qubit. For example, the system may maintain detuning frequencies from a predetermined range of frequencies, e.g., frequencies within a 100 MHz range, such as between 500 MHz and 400 MHz, or within a 200 MHz range, such as between 700 MHz and 500 MHz. If the system pairs multiple data qubits with respective neighboring measurement qubits into multiple non-overlapping subsets of paired data and measurement qubits, the system may perform, for each of the multiple subsets, an entanglement operation nearly in parallel on each paired data and measurement qubit in the subset.

[0117] In some implementations, the system may perform an entanglement operation on each paired data and measurement qubit by applying an entanglement operation frequency trajectory to the paired data and measurement qubit. Exemplary control-Z quantum gate frequency trajectories that may be applied to one or more paired data and respective neighboring measurement qubits are shown with reference to FIG.

[0118] In some implementations, the system may apply respective entanglement operating frequency trajectories to different paired data and measurement qubits. In these implementations, variations between the respective entanglement operating frequency trajectories may be maintained below a predetermined threshold. Such variations may occur due to, for example, variations in control pulse amplitude, as filtered out by the excitation drive lines described herein with reference to FIG. 1.

[0119] 6 shows an exemplary pairing of data and measurement qubits for performing entanglement operations on a first two-dimensional array of qubits 600 and a second two-dimensional array of qubits 650. Both two-dimensional arrays of qubits 600 and 650 include a plurality of data qubits, e.g., data qubits 614 and 616, and a plurality of measurement qubits, e.g., measurement qubits 618 and 620. Each data qubit of the plurality of data qubits is coupled to a plurality of neighboring measurement qubits via a respective qubit coupler, as described herein with reference to FIG.

[0120] Each paired data and neighboring measurement qubit in two-dimensional array 600 does not overlap with another paired data and neighboring measurement qubit. In addition, each paired data and neighboring measurement qubit has a parallel north-south qubit coupler in the same direction, i.e., each measurement qubit is coupled to a data qubit from the south. For convenience, the couplers in each paired data and measurement qubit are shown as north-south couplers, but the couplers could also be south-north couplers (where each measurement qubit is coupled to a data qubit from the north), east-west couplers (where each measurement qubit is coupled to a data qubit from the west), or west-east couplers (where each measurement qubit is coupled to a data qubit from the east).

[0121] A first exemplary two-dimensional array of qubits 600 includes three non-overlapping subsets. Each subset includes multiple paired data and neighboring measurement qubits. Referring to FIG. 6 , the first subset includes all qubits encapsulated by the solid lines, including, for example, qubit pairs 602, 624, and 622. The second subset includes all qubits encapsulated by the thick dashed lines, including, for example, qubit pair 604. The third subset includes all qubits encapsulated by the thin dashed lines, including, for example, qubit pair 606. In some cases, the pairing of data and neighboring measurement qubits as shown in qubit array 600 may not be exhaustive. For example, some qubits at the boundaries of the grid may not be paired with other qubits.

[0122] Each subset includes non-adjacent pairs of data qubits and neighboring measurement qubits, where a qubit is said to be adjacent to another qubit if it is coupled to or diagonal to the other qubit. That is, pairs in each subset are not adjacent to other pairs in the subset. Thus, when entangling operations are performed substantially in parallel on the data and neighboring measurement qubits of each pair in each subset, each qubit involved in each entangling operation is not adjacent to the other qubits involved in each other entangling operation. For example, when entangling operations are performed in parallel on pairs included in the subsets represented by solid lines, the measurement qubit in pair 622 can vary its frequency without crossing the resonance of another measurement qubit that is executing a similar frequency trajectory, because the measurement qubit diagonal to the measurement qubit in pair 622 is a member of the other subsets represented by the thick and thin dashed lines. As explained above, this configuration reduces the probability of parasitic occupation qubit leakage.

[0123] Each paired data and measurement qubit in two-dimensional array 650 has parallel qubit couplers of different orientations: an East-West coupler (where each measurement qubit is coupled to a data qubit from the west) or a West-East coupler (where each measurement qubit is coupled to a data qubit from the east). In other words, each measurement qubit in array 650 can be coupled to a data qubit via an East-West coupler, a West-East coupler, or both. Similarly, each data qubit in array 650 can be coupled to a measurement qubit via an East-West coupler, a West-East coupler, or both. For convenience, the couplers in each paired data and measurement qubit are shown as East-West and West-East couplers, but the couplers could also be North-South and North-South couplers.

[0124] In some implementations, the pattern described above may be further repeated using north-south or north-south couplers to allow all nearest-neighbor data qubit and measurement qubit pairs to undergo the interaction.

[0125] The second exemplary two-dimensional array of qubits 650 includes four overlapping subsets. Each subset includes multiple paired data and neighboring measurement qubits. Referring to FIG. 6 , the first subset includes all qubits encapsulated by solid lines, including, for example, qubit pair 654. The second subset includes all qubits encapsulated by thick dashed lines, such as, for example, qubit pair 658. The third subset includes all qubits encapsulated by thin dashed lines, such as, for example, qubit pair 656. The fourth subset includes all qubits encapsulated by dotted lines, such as, for example, qubit pair 652. In some cases, as shown in qubit array 650, the pairing of data and measurement qubits may be exhaustive, i.e., each qubit may be paired with at least one other qubit.

[0126] Each subset includes adjacent pairs of data qubits and neighboring measurement qubits. For example, a data qubit in each subset is either diagonal to at least one other data qubit in the subset, or a measurement qubit in each subset is diagonal to at least one other measurement qubit in the subset. Thus, when entanglement operations are performed substantially in parallel on each pair of data and neighboring measurement qubits in each subset, each qubit involved in each entanglement operation is adjacent to (diagonal to) another qubit of the same type involved in each other entanglement operation. However, by detuning each measurement qubit in the subset in parallel, e.g., by maintaining a substantially constant detuning Δf between measurement qubits in paired data and measurement qubits, each measurement qubit can be detuned without crossing the resonance of another measurement qubit that is executing a similar frequency trajectory on its corresponding data qubit. As explained above, this configuration reduces the probability of parasitic occupied qubit leakage.

[0127] 7 is a plot 700 of an exemplary controlled-Z quantum gate frequency trajectory 702. Plot 700 shows an exemplary normalized time versus control frequency amplitude (ΔH) during application of an adiabatic controlled-Z quantum gate, as described above with reference to FIG. Z ) For example, the control frequency amplitude may represent the amplitude of the control pulse for the controlled Z quantum gate, as generated by excitation pulse generator 110 and emitted by excitation drive line 124 of FIG. 1 above.

[0128] The exemplary frequency trajectory 702 may be applied to paired data and measurement qubits to perform entanglement operations, e.g., a controlled Z quantum gate. As described above with reference to FIG. 5, in some implementations, the frequency trajectories applied in approximately parallel to each pair of data and measurement qubits may include variations in control pulse amplitude. For example, the control frequency amplitude (ΔH Z ) may be varied, for example, by a factor of 100 MHz, up to that shown in plot 700.

[0129] FIG. 8 is an example plot 800 of the probability of parasitic population leakage versus diagonal coupling strength when performing entanglement operations in parallel on paired data and measurement qubits, as described above with reference to FIG. 5.

[0130] The probability of parasitic occupied qubit leakage during standard entanglement operations on paired data and measurement qubits, e.g., entanglement operations different from those described in this disclosure, can be estimated using the framework of the Landau-Zener transition, which is described, for example, in the framework of "Fast adiabatic qubit gates using only σ Z control,” J. Martinis and M. Geller, Phys. Rev. A 90, 022307 (2014), the disclosure of which is incorporated herein by reference in its entirety. Within this framework, the probability of occupancy leakage can be given by the following equation:

number

number

[0131]

number

number

number

[0132] The probability P of parasitic population leakage during entanglement operations as described herein can be estimated as:

number

number

number

[0133] Using this framework, the probability of parasitic population leakage during entanglement operations according to the present disclosure is plotted as a function of parasitic coupling strength in plot 800. In plot 800, the detuning frequency Δf is varied from 500 MHz to 400 MHz, e.g., due to variations in control pulse amplitude between qubits, as described above with reference to FIG. 5, and η=200 MHz. Plot 800 shows that the probability of parasitic population leakage is 10 5 and 10 7 The diagonal bond strength g between diag / 2π(MHz), <10 -11 This leads to a clear improvement in the probability of crossing the energy levels of diagonal qubits, about 10 9 This will result in improvements in:

[0134] Hardware programming: Surface code cycle In some settings, quantum computers can provide a means for efficiently solving some problems that may not be solved efficiently using conventional classical computers. Example problems include factoring extremely large numbers into their prime numbers and searching large, unstructured data sets. However, physical systems such as systems of ions, spins in semiconductors, and superconducting circuits may not always perform well enough to serve directly as computational qubits in quantum computing devices.

[0135] One approach to building quantum computing devices is based on surface codes. Surface codes provide an error-tolerant way to represent information in quantum computing devices. Logical qubits are constructed from a collection of physical qubits in such a way that the logical qubits can perform better than individual physical qubits.

[0136] In some cases, the surface code can be operated as a stabilizer code, a way for the stabilizers to be measured to detect errors as they occur. By selecting suitable choices of stabilizer measurements, the qubits can be operated to perform logical operations. Measuring the stabilizers across a system of qubits thus constitutes the fundamental iterative cycle for a quantum computer, upon which all higher functions can be built.

[0137] FIG. 9 shows an exemplary quantum circuit 900 for measuring a stabilizer during a surface code error detection cycle. The exemplary quantum circuit 900 includes a five-qubit register. The five-qubit register includes a measurement qubit represented as |0> and four data qubits representing the measurement qubit's nearest neighbors. In the exemplary quantum circuit, the measurement qubit is assumed to be located in a two-dimensional grid, as described with reference to FIG. 1 . Thus, the four nearest neighbor data qubits correspond to the data qubit |S> from the south, the data qubit |W> from the west, the data qubit |E> from the east, and the data qubit |N> from the north.

[0138] In some cases, for example, if a measurement qubit has fewer neighboring data qubits, the quantum circuit may have a smaller qubit register. For example, if a measurement qubit is at a corner of a two-dimensional grid, the measurement qubit may have only two neighboring data qubits. In this example, the corresponding quantum circuit may have three qubit registers.

[0139] Exemplary quantum circuit 900 illustrates the sequence of quantum logic gates required to perform a surface code error detection cycle 1000, as described herein with reference to FIG. 10. As described herein with reference to FIG. 10, exemplary quantum circuit 900 includes a first Hadamard gate 952 applied to the measurement qubit |0>. A first entanglement operation 956 is then performed on the measurement qubit register |0> and the data qubit register |S> from the south. A second Hadamard gate 958 is then applied to the data qubit register |W> from the west. A second entanglement operation 960 is then applied to the measurement qubit register |0> and the data qubit register |W> from the west.

[0140] Exemplary quantum circuit 900 includes third and fourth Hadamard gates 972 and 974. Hadamard gates 972 and 974 are sequentially applied to the qubit |W〉 from the west and the qubit |E〉 from the east, respectively. When exemplary quantum circuit 900 is applied to a system of measurement qubits and data qubits, such as described above with reference to FIG. 1 , the Hadamard gate applied to the data qubit from the west (after the entanglement operation between the first measurement qubit and the data qubit from the west) is canceled by the Hadamard gate applied to the data qubit from the east (before the entanglement operation between the second measurement qubit and the data qubit from the east).

[0141] A third entanglement operation 962 is applied to the measurement qubit register |0> and the data qubit register |E> from the East. Then, a fifth Hadamard gate 964 is applied to the data qubit register |E> from the East. A fourth entanglement operation 966 is applied to the measurement qubit register |0> and the data qubit register |N> from the North. A sixth Hadamard gate 968 is applied to the measurement qubit register |0>, followed by a measurement operation 970.

[0142] Entangling operations 956, 960, 962, and 966 may include controlled Z quantum logic gates. For example, when Hadamard quantum logic gates, such as Hadamard quantum logic gates 958 and 972, or 974 and 964, are applied before and after a controlled Z quantum logic gate, the three gates together operate as a (Hadamard, controlled Z, Hadamard) controlled X quantum logic gate. Thus, taken together, the entanglement operations shown in FIG. 9 may represent the application of operators ZXXZ (control Z, control X, control X, control Z) when the measurement qubit is in the |1> state.

[0143] 10 is a flow diagram of an exemplary process 1000 for performing surface code error detection cycles on multiple quantum circuits, such as the quantum circuit shown in FIG. 9. For convenience, process 1000 is described as being performed by a system of one or more quantum or classical computers located at one or more locations. For example, process 1000 may be implemented using qubit controller 104 of system 100 described above with reference to FIG. 1. In some implementations, process 1000 may be performed in conjunction with the techniques described above with reference to FIGS. 2-8.

[0144] The exemplary process 1000 is described as being performed by a system on a plurality of data qubits and a plurality of measurement qubits arranged as a two-dimensional grid, for example, grid 112 of FIG. 1, where each data qubit of the plurality of data qubits is coupled to a neighboring measurement qubit via a respective qubit coupler, as described above with reference to FIG. 1.

[0145] The system initializes a plurality of measurement qubits (step 1002). For example, as shown in exemplary quantum circuit 900 of FIG. 9, initializing the plurality of measurement qubits may include preparing the measurement qubits in a |0> computational basis state.

[0146] The system applies a Hadamard quantum logic gate to the initialized measurement qubit (step 1004). Initializing the measurement qubit in a |0> computational basis state and applying a Hadamard quantum logic gate to the initialized measurement qubit places the measurement qubit in a 50 / 50 superposition of |0> and |1>. Application of a Hadamard quantum logic gate 952 to the initialized measurement qubit is shown above with reference to FIG. 9.

[0147] The system performs multiple entanglement operations on the first set of paired measurement and data qubits (step 1006). For example, the entanglement operations may include a controlled-Z quantum logic gate. Applying the controlled-Z quantum logic gate to the paired measurement and data qubits includes applying a Z operator to the data qubit when the measurement qubit is in state |1>.

[0148] Each pair in the first set of paired measurement and data qubits includes a measurement qubit coupled to a neighboring data qubit in a first direction. For example, each pair may include a measurement qubit coupled via a respective qubit coupler to a neighboring data qubit below the measurement qubit, e.g., in a southward direction. Exemplary pairs of measurement qubits coupled to respective neighboring data qubits in the southward direction are shown and described herein with reference to two-dimensional qubit grid 600 of FIG. 6. Application of entanglement operation 956 to a measurement qubit paired with a southward data qubit is shown with reference to FIG. 9.

[0149] In some implementations, performing the multiple entanglement operations on the first set of paired measurement and data qubits includes separating the paired measurement and data qubits into multiple subsets of paired qubits, where the multiple subsets include overlapping and non-adjacent pairs. In these implementations, non-adjacent is understood to include diagonally non-adjacent pairs. An exemplary multiple subsets of paired qubits is shown and described above with reference to the two-dimensional qubit grid 600 of FIG. 6. As shown in the two-dimensional qubit grid 600 of FIG. 6, in some implementations, the multiple subsets may include three subsets 602, 604, and 606.

[0150] The system may then perform an entanglement operation in parallel on pairs of qubits in each of the multiple subsets. For example, as described with reference to FIG. 5, performing an entanglement operation in parallel on pairs of qubits in each of the multiple subsets may include detuning each measurement qubit in each subset in parallel.

[0151] The system applies a Hadamard quantum logic gate to the plurality of data qubits in a second direction (step 1008). For example, the system may apply a Hadamard quantum logic gate to the plurality of data qubits in a direction from the west of the measurement qubit. The application of a Hadamard quantum logic gate 958 to the data qubits from the west is shown with reference to FIG. 9.

[0152] The system performs multiple entanglement operations on the second set of paired measurement and data qubits (step 1010). The operations may include controlled Z quantum logic gates and Hadamard quantum logic gates. For example, the system may perform a controlled Z quantum logic gate on the measurement qubit paired with the data qubit in the second direction, followed by a Hadamard quantum logic gate on the data qubit in the second direction. The system may then perform a Hadamard quantum logic gate on the data qubit in the third direction, followed by a controlled Z quantum gate on the measurement qubit paired with the data qubit in the third direction.

[0153] Each pair in the second set of paired measurement and data qubits includes a measurement qubit coupled to a neighboring data qubit in a second or third direction, where the second and third directions are orthogonal to the first direction and the second direction is opposite to the third direction. For example, each pair may include a measurement qubit coupled via a respective qubit coupler to a neighboring data qubit to the right or left of the measurement qubit, i.e., in the east or west direction. Because entangling operations from the west and the east are interchangeable, the system may perform a mixture of entangling operations from the west and the east.

[0154] Exemplary pairs of measurement qubits coupled to respective neighboring data qubits in the east and west directions are shown and described above with reference to two-dimensional qubit grid 650 of Figure 6. The application of entanglement operations 960 and 962 applied to measurement qubits paired with data qubits from the west and east, respectively, is shown with reference to Figure 9.

[0155] In some implementations, performing the multiple entanglement operations on the second set of paired measurement and data qubits includes separating the paired measurement and data qubits into multiple subsets of paired qubits, where the multiple subsets include overlapping and adjacent pairs. In these implementations, adjacent is understood to include diagonally adjacent pairs. Exemplary multiple subsets of such paired qubits are shown and described above with reference to the two-dimensional qubit grid 650 of FIG. 6. As shown in the two-dimensional qubit grid 650 of FIG. 6, in some implementations, the multiple subsets may include four subsets 652, 654, 656, and 658.

[0156] The system may then perform an operation in parallel on pairs of qubits in each of the multiple subsets. For example, as described with reference to Figure 5, performing an entanglement operation in parallel on pairs of qubits in each of the multiple subsets may include detuning each measurement qubit in each subset in parallel.

[0157] The system applies a Hadamard quantum logic gate to the plurality of data qubits in a third direction (step 1012). For example, the system may apply a Hadamard quantum logic gate to the plurality of data qubits in a direction east from the measurement qubit. The application of a Hadamard quantum logic gate 964 to the data qubits from the east is illustrated with reference to FIG. 9. As described with reference to FIG. 9, when a Hadamard quantum logic gate is applied before and after a controlled Z quantum logic gate, for example, as described with reference to steps 1010 and 1012, the three gates together act as a controlled X quantum logic gate.

[0158] The system performs multiple entanglement operations on the third set of paired measurement and data qubits (step 1014). As described above, the entanglement operations may include controlled Z quantum logic gates. Each pair in the third set of paired measurement and data qubits includes a measurement qubit coupled to a neighboring data qubit in a fourth direction, the fourth direction opposite the first direction. For example, each pair may include a measurement qubit coupled via a respective qubit coupler to a neighboring data qubit above the measurement qubit, i.e., in the north direction. Exemplary pairs of measurement qubits coupled to respective neighboring data qubits in the north direction can be obtained from a simple modification of the two-dimensional qubit grid 600 of FIG. 6. Application of the entanglement operation 966 to a measurement qubit paired with a north data qubit is illustrated with reference to FIG. 9.

[0159] In some implementations, performing multiple entanglement operations on the third set of paired measurement and data qubits includes separating the paired measurement and data qubits into multiple subsets of paired qubits, where the multiple subsets include overlapping and non-adjacent pairs. In these implementations, non-adjacent is understood to include diagonally non-adjacent pairs. In some implementations, the multiple subsets may include three subsets.

[0160] The system may then perform an entanglement operation in parallel on pairs of qubits in each of the multiple subsets. For example, as described with reference to FIG. 5, performing an entanglement operation in parallel on pairs of qubits in each of the multiple subsets may include detuning each measurement qubit in each subset in parallel.

[0161] The system applies a Hadamard quantum logic gate to the plurality of measurement qubits, step 1016. The application of a Hadamard quantum logic gate 968 to the measurement qubits is illustrated with reference to FIG.

[0162] The system measures the plurality of measurement qubits to detect errors (step 1018). An exemplary measurement operation 970 is shown with reference to FIG.

[0163] As explained above, performing multiple entanglement operations on a first set of paired measurement and data qubits requires three sequential applications of the array of entanglement operations, one application per subset. If this scheme were applied individually for all four directions of nearest-neighbor interactions, e.g., north, south, east, and west, a complete surface code error detection cycle 1000 would require 12 applications of entanglement operations. However, by detuning geometrically diagonal measurement qubits in parallel, e.g., according to the techniques described above with reference to Figures 5 and 6, a denser pattern of entanglement operations allows all interactions orthogonal to the first direction, e.g., east and west, to be completed in only four layers of CZ gates, one application per subset, for a total of only 10 applications of entanglement operations.

[0164] In some cases, the system may further perform leakage elimination. For example, the system may perform leakage elimination in parallel with the final entanglement operation of each measurement qubit, e.g., in parallel with step 1014 described above. For example, the system may swap measurement qubits and data qubits such that each type of qubit is alternatively reset. This may be achieved by applying controlled Z+ swap quantum logic gates that interact and transfer information in the computational basis states |0> and |1> but do not transfer information in states |2> and above.

[0165] In some implementations, subsequent surface code error detection cycles may be performed in the reverse order of the cycles described in steps 1002-1018 above. For example, rather than performing a south-west / east-north detection cycle as described above, the system may perform a north-west / east-south detection cycle. That is, the system may initialize a plurality of measurement qubits, apply Hadamard quantum logic gates to the initialized measurement qubits, perform entanglement operations in parallel on a third subset of paired data and measurement qubits, apply Hadamard quantum logic gates to a plurality of data qubits, perform entanglement operations in parallel on a second subset of paired data and measurement qubits, apply Hadamard quantum logic gates to the plurality of data qubits, perform entanglement operations in parallel on a first subset of paired data and measurement qubits, apply Hadamard quantum logic gates to the plurality of measurement qubits, and measure the plurality of measurement qubits to detect errors. By performing subsequent surface code error detection cycles in this order, it can be ensured that the data remains local, e.g., the information read out from each paired measurement qubit corresponds only to its respective data qubit.

[0166] FIG. 11 shows an example implementation 1100 of a surface code. The example implementation 1100 shows a two-dimensional array of qubits, as described above with reference to FIG. 1. Each of the qubits in the two-dimensional array of qubits is represented as an open circle, e.g., qubit 1104, or a filled circle, e.g., 1106. In some implementations, the open circles represent data qubits, as described above with reference to FIG. 1. In these implementations, the filled circles represent measurement qubits, as described above with reference to FIG. 1. For clarity, the two-dimensional array of qubits includes 5×5 qubits, although in some cases, surface code implementations may include fewer or more qubits.

[0167] As described above with reference to FIG. 1 , the qubits interact with each other via multiple nearest-neighbor qubit couplers, which for convenience are not shown in exemplary implementation 1100. Thus, away from the array boundary, each data qubit contacts four measurement qubits, and each measurement qubit contacts four data qubits. Thus, a measurement qubit performs four measurements. At the array boundary, a measurement qubit contacts three data qubits and performs three measurements, and a data qubit contacts either two or three measurement qubits.

[0168] The exemplary implementation 1100 includes multiple uniform stabilizers, e.g., stabilizer 1102. The stabilizers are used to maintain the quantum state of an array of qubits. By repeatedly measuring a quantum system using a complete set of stabilizers, which are generally interchangeable, the quantum system is forced into simultaneous and unique eigenstates of all stabilizers. The stabilizers can be measured without perturbing the system. When the measurement result changes, this corresponds to one or more qubit errors, and the quantum state is projected into a different stabilizer eigenstate by the measurement. Surface code stabilizers are described, for example, in “Surface codes: Towards practical large-scale quantum computation,” A. Fowler et al., Phys. Rev. A 86, 032324 (2012), the disclosure of which is incorporated herein by reference in its entirety.

[0169] Each stabilizer in the example implementation 1102 includes:

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[0170] Embodiments of the digital and / or quantum subject matter, and digital functional operations and quantum operations described herein, including the structures disclosed herein and their structural equivalents, may be implemented in digital electronic circuitry, suitable quantum circuitry, or more generally, in a quantum computing system, in tangibly embodied digital and / or quantum computer software or firmware, in digital and / or quantum computer hardware, or in one or more combinations thereof. The term "quantum computing system" may include, but is not limited to, a quantum computer, a quantum information processing system, a quantum cryptography system, or a quantum simulator.

[0171] Embodiments of the digital and / or quantum subject matter described herein may be implemented as one or more digital and / or quantum computer programs, i.e., one or more modules of digital and / or quantum computer program instructions encoded on a tangible, non-transitory storage medium for execution by or to control the operation of a data processing apparatus. The digital and / or quantum computer storage medium may be a machine-readable storage device, a machine-readable storage substrate, a random or sequential access memory device, one or more qubits, or a combination of one or more thereof. Alternatively or additionally, the program instructions may be encoded on an artificially generated propagated signal capable of encoding digital and / or quantum information, e.g., a machine-generated electrical, optical, or electromagnetic signal generated to encode digital and / or quantum information for transmission to a receiver device suitable for execution by a data processing apparatus.

[0172] The terms quantum information and quantum data refer to information or data carried by or held or stored in a quantum system, where the smallest nontrivial system is a qubit, i.e., a system defining a unit of quantum information. It will be understood that the term "qubit" encompasses all quantum systems that can be suitably approximated as a two-level system in the corresponding context. Such quantum systems may include multi-level systems, e.g., with more than two levels. By way of example, such systems may include atoms, electrons, photons, ions, or superconducting qubits. In many implementations, the computational basis state is identified using the ground state and a first excited state, although it will be understood that other setups are possible in which the computational state is identified using a higher-level excited state.

[0173] The term "data processing apparatus" refers to digital and / or quantum data processing hardware and encompasses all types of apparatus, devices, and machines for processing digital and / or quantum data, including, by way of example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, multiple digital and quantum processors or computers, and combinations thereof. An apparatus may also be or further include special-purpose logic circuitry, e.g., an FPGA (field-programmable gate array), an ASIC (application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing apparatus designed to simulate or create information about a particular quantum system. Specifically, a quantum simulator is a special-purpose quantum computer that does not have the capability to perform universal quantum computation. In some cases, an apparatus may include, in addition to hardware, code that creates an execution environment for digital and / or quantum computer programs, e.g., code constituting processor firmware, a protocol stack, a database management system, an operating system, or one or more combinations thereof.

[0174] A digital computer program, which may be called or described as a program, software, software application, module, software module, script, or code, may be written in any type of programming language, including a compiled or interpreted language, or a declarative or procedural language, and may be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program, which may be called or described as a program, software, software application, module, software module, script, or code, may be written in any type of programming language, including a compiled or interpreted language, or a declarative or procedural language, and may be converted to a suitable quantum programming language or written in a quantum programming language, e.g., QCL or Quipper.

[0175] A digital and / or quantum computer program may, but need not, correspond to a file in a file system. A program may be stored in a portion of a file holding other programs or data, e.g., one or more scripts stored in a markup language document, in a single file dedicated to the program in question, or in multiple coordinated files, e.g., files storing one or more modules, subprograms, or portions of code. A digital and / or quantum computer program may be deployed to run on one digital or quantum computer, or on multiple digital and / or quantum computers located at one site or distributed across multiple sites and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that can transmit quantum data using quantum systems, e.g., qubits. Generally, digital data communication networks cannot transmit quantum data, but quantum data communication networks can transmit both quantum data and digital data.

[0176] The processes and logic flows described herein may be performed by one or more programmable digital and / or quantum computers operating in conjunction with one or more digital and / or quantum processors, executing one or more digital and / or quantum computer programs, as appropriate, to perform functions by operating on input digital and quantum data and generating output. The processes and logic flows may also be performed by, and an apparatus may be implemented as, special purpose logic circuitry, e.g., an FPGA or ASIC, or a quantum simulator, or a combination of special purpose logic circuitry or a quantum simulator with one or more programmed digital and / or quantum computers.

[0177] One or more digital and / or quantum computer systems "configured to" perform particular operations or actions means that the system has installed thereon software, firmware, hardware, or a combination thereof that causes the system to perform the operations or actions during operation. One or more digital and / or quantum computer programs configured to perform particular operations or actions means that the one or more programs contain instructions that, when executed by a digital and / or quantum data processing device, cause the device to perform the operations or actions. A quantum computer may receive instructions from a digital computer that, when executed by a quantum computing device, cause the device to perform the operations or actions.

[0178] A digital and / or quantum computer suitable for executing a digital and / or quantum computer program may be based on a general-purpose or a dedicated digital and / or quantum processor, or both, or any other kind of central digital and / or quantum processing unit. Generally, the central digital and / or quantum processing unit will receive instructions and digital and / or quantum data from a read-only memory, a random access memory, or a quantum system suitable for transmitting quantum data, e.g., photons, or a combination thereof.

[0179] The essential elements of a digital and / or quantum computer are a central processing unit for carrying out or executing instructions and one or more memory devices for storing instructions and digital and / or quantum data. The central processing unit and memory may be supplemented by or incorporated into dedicated logic circuits or quantum simulators. Generally, a digital and / or quantum computer will also include one or more mass storage devices for storing digital and / or quantum data, such as magnetic disks, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information, or be operatively coupled to receive digital and / or quantum data therefrom, transfer digital and / or quantum data thereto, or both. However, a digital and / or quantum computer need not have such devices.

[0180] Digital and / or quantum computer-readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include, by way of example, all forms of non-volatile digital and / or quantum memories, media, and memory devices, including semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices, magnetic disks, e.g., internal hard disks or removable disks, magneto-optical disks, CD-ROM and DVD-ROM disks, and quantum systems, e.g., trapped atoms or electrons. It will be understood that a quantum memory is a device capable of storing quantum data with high fidelity and efficiency for long periods of time, e.g., a light-matter interface where light is used for transmission and material for storing and preserving quantum characteristics of quantum data, such as superposition or quantum coherence.

[0181] Control of various systems described herein, or portions thereof, may be implemented in a digital and / or quantum computer program product that includes instructions stored on one or more non-transitory machine-readable storage media and executable on one or more digital and / or quantum processing devices. The systems described herein, or portions thereof, may be implemented as apparatuses, methods, or systems, each of which may include one or more digital and / or quantum processing devices and a memory for storing executable instructions for performing the operations described herein.

[0182] While the specification contains numerous specific implementation details, these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be inherent in particular embodiments. Some features described herein in the context of separate embodiments may also be implemented in combination in a single embodiment. Conversely, various features described in the context of a single embodiment may also be implemented in multiple embodiments separately or in any suitable subcombination. Moreover, although features may be described above as operative in some combinations, and may even initially be claimed as such, one or more features from a claimed combination may, in some cases, be deleted from the combination, and the claimed combination may be directed to a subcombination or a variation of a subcombination.

[0183] Similarly, while operations are illustrated in the figures in a particular order, this should not be understood as requiring such operations to be performed in the particular order illustrated, or in sequential order, or that all of the illustrated operations be performed to achieve desirable results. In some situations, multitasking and parallel processing may be advantageous. Moreover, the separation of various system modules and components in the above-described embodiments should not be understood as requiring such separation in all embodiments, and it should be understood that the described program components and systems may generally be integrated together in a single software product or packaged in multiple software products.

[0184] Specific embodiments of the subject matter have been described. Other embodiments are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes illustrated in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous. [Explanation of symbols]

[0185] 100 Quantum Computing System, System 102, 116, 118, 610, 612, 1104, 1106 qubits 104 Qubit Controller Module, Qubit Controller 106 Entanglement operating frequency 108 qubit operating frequency 110 Excitation Pulse Generator 112 2D Grid, Grid 114 Qubit Coupler 120 Parasitic Coupling 122 Qubit Frequency Control Line 124 Excitation Drive Line 126 Combiner 300, 350 Schematic data qubit frequency pattern 302 data qubit, first data qubit, qubit 304 qubit, second data qubit, data qubit 306 third data qubit, data qubit, qubit 308 Fifth Data Qubit, Data Qubit 310 fourth data qubit, data qubit 352, 354, 356, 358, 360, 614, 616 Data Qubits 400 Data Qubit Frequency and Measurement Qubit Frequency, Example 402 Data Qubit Frequency 404 Interaction Frequency 406 Measuring Qubit Frequency 408 Read and Reset Frequency 410, 412, 414 frequency range 416 additional read and reset frequencies 600 two-dimensional qubit grid, two-dimensional grid, first two-dimensional array of qubits, two-dimensional array of qubits, two-dimensional array, first exemplary two-dimensional array of qubits, qubit array 602 Groups, Qubit Pairs, and Subsets 604, 606, 652, 654, 656, 658 qubit pairs, subsets 608 Group 624 qubit pairs 618, 620 Measurement qubit 622 qubit pairs, pairs 626, 628, 630 Arrows 650 2D grid, second 2D array of qubits, 2D array of qubits, 2D array, array, second exemplary 2D array of qubits, qubit array, 2D qubit grid 700, 800 plots 702 Controlled Z quantum gate frequency trajectory, frequency trajectory 900 Exemplary Quantum Circuits 952 First Hadamard Gate, Hadamard Quantum Logic Gate 956 First Entanglement Action, Entanglement Action 958 Second Hadamard Gate, Hadamard Quantum Logic Gate 960 Second Entanglement Operation, Entanglement Operation 962 Third Entanglement Action, Entanglement Action 964 Fifth Hadamard Gate, Hadamard Quantum Logic Gate 966 Fourth Entanglement Action, Entanglement Action 968 Sixth Hadamard Gate, Hadamard Quantum Logic Gate 970 Measurement Operation 972 Third Hadamard Gate, Hadamard Gate, Hadamard Quantum Logic Gate 974 Fourth Hadamard Gate, Hadamard Gate, Hadamard Quantum Logic Gate 1000 bit error detection cycles, process 1100 Surface Code Example Implementations, Example Implementations 1102 stabilizer

Claims

1. operating a first qubit in a system of qubits at a first qubit frequency within a first qubit frequency range, the first qubit frequency range covering a first frequency range; operating a second qubit in the system of qubits at a second qubit frequency within a second qubit frequency range, the second qubit frequency range covering a second frequency range, the second qubit frequency and the second qubit frequency range being different from the first qubit frequency and the first qubit frequency range, the first qubit frequency differing from the second qubit frequency by at least 2η, where η represents a system nonlinearity, and wherein the first qubit and second qubit pair are diagonal to one qubit in a two-dimensional grid; operating a third qubit in the system of qubits at a third qubit frequency derived from the first qubit frequency range, the third qubit frequency being different from the first qubit frequency and the second qubit frequency; the third qubit is different from the second qubit, and the third qubit frequency differs from the second qubit frequency by at least 2η, where η represents a system nonlinearity, and wherein the second qubit and third qubit pair is diagonal to one qubit in a two-dimensional grid; A method comprising:

2. the first qubit frequency and the second qubit frequency differ by 2η, where η represents a system nonlinearity; the first qubit frequency and the third qubit frequency differ by 2η, where η represents a system nonlinearity; The method of claim 1.

3. 2. The method of claim 1, wherein the nonlinearity of the system of qubits is equal to 0.2 GHz.

4. the first qubit frequency is a first data qubit frequency; the second qubit frequency is a second data qubit frequency; the first qubit is a first data qubit and the second qubit is a second data qubit; the first qubit frequency range comprises a first data qubit frequency range; the second qubit frequency range comprises a second data qubit frequency range; The method of claim 1.

5. The first qubit frequency range and the second qubit frequency range are: Idling frequency range, Echo operating frequency domain, Single-qubit gate frequency domain, or interaction frequency domain and The method of claim 1.

6. 2. The method of claim 1, wherein a difference between the second qubit frequency and the third qubit frequency is greater than a next-nearest neighbor coupling strength of a coupling between the second qubit and the third qubit.

7. The method of claim 6 , wherein the next nearest neighbor coupling strength is equal to 1 MHz.

8. 10. The method of claim 1, wherein the second qubit frequency range comprises a width of 10 MHz.

9. 1. An apparatus including a system of qubits, the system of qubits comprising: a first qubit at a first qubit frequency within a first qubit frequency range, the first qubit frequency range covering a first frequency range; a second qubit at a second qubit frequency within a second qubit frequency range, the second qubit frequency range covering a second frequency range, the second qubit frequency and the second qubit frequency range being different from the first qubit frequency and the first qubit frequency range, the first qubit frequency differing from the second qubit frequency by at least 2η, where η represents a system nonlinearity, and wherein the pair of the first qubit and the second qubit is diagonal to one qubit in a two-dimensional grid; a third qubit at a third qubit frequency from the first qubit frequency range, the third qubit frequency different from the first qubit frequency and the second qubit frequency, the third qubit different from the second qubit, and the third qubit frequency differs from the second qubit frequency by at least 2η, where η represents a system nonlinearity, and wherein the second qubit and third qubit pair is diagonal to one qubit in a two-dimensional grid; Device.

10. 10. The apparatus of claim 9, further comprising a qubit controller module configured to operate the system of qubits, the qubit controller module comprising an excitation pulse generator and one or more excitation drive lines, the qubit controller module controlling the qubits via excitation pulses on the excitation drive lines to operate the qubits at a qubit frequency derived from a qubit frequency range.

11. The apparatus of claim 10 , wherein the one or more excitation drive lines comprise a global excitation drive line.

12. the first qubit frequency and the second qubit frequency differ by 2η, where η represents a system nonlinearity; the first qubit frequency and the third qubit frequency differ by 2η, where η represents a system nonlinearity; 10. The apparatus of claim 9.

13. 10. The apparatus of claim 9, wherein the nonlinearity of the system of qubits is equal to 0.2 GHz.

14. the first qubit frequency is a first data qubit frequency; the second qubit frequency is a second data qubit frequency; the first qubit is a first data qubit and the second qubit is a second data qubit; the first qubit frequency range comprises a first data qubit frequency range; the second qubit frequency range comprises a second data qubit frequency range; 10. The apparatus of claim 9.

15. The first qubit frequency range and the second qubit frequency range are: Idling frequency range, Echo operating frequency domain, Single-qubit gate frequency domain, or interaction frequency domain and 10. The apparatus of claim 9.

16. 10. The apparatus of claim 9, wherein a difference between the second qubit frequency and the third qubit frequency is greater than a next-nearest neighbor coupling strength of a coupling between the second qubit and the third qubit.

17. 17. The apparatus of claim 16, wherein the next nearest neighbor coupling strength is equal to 1 MHz.

18. 10. The apparatus of claim 9, wherein the second qubit frequency range comprises a width of 10 MHz.

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