Method for measuring observable quantity of cat quantum bits

Abstract

The invention relates to a method for measuring observable quantity of cat quantum bits. A quantum system includes a command circuit and a non-linear superconducting quantum circuit including a non-linear element and a resonant portion. The non-linear superconducting quantum circuit has first and second modes with respective resonant frequency sums. The method comprises: a) delivering (900), by the command circuit, microwave radiation at a frequency to drive the second mode so that the nonlinear element constructs a hopping operator that stabilizes a two-dimensional manifold carrying the cat qubits, b) mapping (910) the observable quantity to be measured to a Fouli operator Z in a cat qubit manifold having a square modulus of magnitude greater than or equal to 2, and c) measuring (920) the value of the Fouli operator Z.

Description

Technical Field

[0001] The field of this invention relates to the measurement of observable quantities of cat qubits. More specifically, the observable quantities to be measured are Pauli operators X, Y, or parity operators. Background Technology

[0002] In general, superconducting qubits can be implemented as a two-stage system of superconducting electronic circuits. Such qubits can be stored in Boson mode, thus forming a specific class of superconducting qubits known as wave qubits.

[0003] As is known from existing technology, cat qubits can be stabilized, that is, Bose qubits defined by a quantum manifold spanned by the superposition of two coherent states, which are quasi-classical states of the Bose mode. To this end, a specific dissipative stabilization mechanism can be implemented, which involves engineering a nonlinear transition between two photons in a first mode (also known as the storage mode or cat qubit mode) carrying a stable quantum manifold and one photon in a strongly dissipative second mode (also known as the buffer mode).

[0004] R. Lescanne et al. in " Exponential suppression of bit-flips in a qubit encoded in an oscillator The possibility of realizing such a qubit has been demonstrated in Nature Physics, 2020 (Exponential Suppression of Bit Flips in Qubits Encoded in Oscillators). As mentioned above, cat qubits rely on a mechanism of dissipating photons in pairs. This process locks two coherent states to separate locations in phase space. By increasing this interval, experiments have shown an exponential decrease in the bit flip rate, while only a linear increase in the phase flip rate. As a result, stable cat qubits have been shown to benefit from a high-noise bias, meaning that the bit flip probability is exponentially smaller than the phase flip probability.

[0005] To determine the state of a cat qubit, a transmon is typically used as a measurement device coupled to the cat qubit mode. For example, Lescanne et al. (2020) proposed using an asymmetric tunneling superconducting quantum interference device (hereinafter referred to as ATS) to engineer a 2-to-1 photon conversion associated with the transmon used to measure the state of the cat qubit. However, as noted in that paper, such an architecture causes the bit-flipping time to saturate to a few milliseconds. The applicants' work reveals that this is because the quantum manifold stabilization rate is too small to resist the dispersion shift caused by thermal excitation of the measurement device. In fact, the transmon has a parasitic cross-Kerr term, which introduces an additional noisy process with an escape rate outside the quantum manifold given by a very large transmon-cat-qubit dispersion shift.

[0006] Furthermore, the transmon problem has been observed in the first embodiment of the aforementioned stabilization scheme—Z. Leghtas et al. (2015) in their article " Confining the state of light to a quantum manifold by engineered two-photon loss( The state of light is confined to a quantum manifold through engineered two-photon loss. ） (Science, Vol. 347, No. 6224) and S. Touzard et al. (2018) in the article " Coherent Oscillations inside a Quantum Manifold Stabilized by Dissipation (Coherent oscillations within a quantum manifold stabilized by dissipation) (Physical Review X8, 023005) — where the superconducting circuit element used as a four-wave mixer is a transmon with a single Josephson junction.

[0007] Therefore, it is necessary to measure the state of cat qubits without using transmon. This need is addressed by recent work—Berdou et al. (2022) in their paper "..." One hundred second bit-flip time in a two-photon dissipative oscillator The approach, driven by the 100-second bit-flip time in a two-optical dissipative oscillator (arXiv:2204.09128, https: / / arxiv.org / pdf / 2204.09128.pdf), achieves approximately 100 seconds of bit-flip time by removing the measurement transmon and operating the ATS in a state that should be dynamically stable. On the other hand, the measurement device used in Berdou et al. (2022), which replaces the transmon, cannot measure the observables of the cat qubits associated with the quantum superposition of coherent states of the cat qubits. However, measuring such observables is a prerequisite for any quantum application.

[0008] This invention attempts to improve this situation. Summary of the Invention

[0009] To this end, the applicant proposes a method for measuring observables of cat qubits implemented by a quantum system, wherein the observables to be measured are measured between a parity check operator, a Pauli operator X, and a Pauli operator Y.

[0010] The quantum system includes a command circuit for delivering microwave radiation and a nonlinear superconducting quantum circuit. The nonlinear superconducting quantum circuit includes a three-wave or four-wave mixing nonlinear element and at least one resonant section. The three-wave or four-wave mixing nonlinear element is connected to the at least one resonant section. The nonlinear superconducting quantum circuit has a first mode with a first resonant frequency and a second mode with a second resonant frequency.

[0011] This method includes the following operations:

[0012] a) The command circuit delivers microwave radiation at a frequency equal to the second resonant frequency to the at least one resonant portion to drive the second mode, thereby causing the three-wave or four-wave mixing nonlinear element to construct a jump operator. The jump operator Represented as This stabilizes the two-dimensional manifold carrying the cat qubits, wherein It is the two-photon dissipation rate, It is the annihilation operator of the first mode, and It is a complex number produced by the delivered microwave radiation.

[0013] b) Mapping the observable to be measured to the Pauli operator Z, the mapping depending on the observable to be measured, and including at least: b1) turning off or reducing the drive of the second mode to make When the square modulus reaches a value less than 1, and subsequently b2) enable or increase the drive of the second mode to make To achieve a square modulus greater than or equal to 2, the mapping further includes b3) during b1) or b2). Applying a time less than 2 to the cat qubit -Pauli Men, and

[0014] c) Measure the value of the Pauli operator Z.

[0015] According to one or more embodiments, the observable to be measured is the Pauli operator Y, and b3) includes Pauli-X ( A gate is applied to the cat qubit.

[0016] According to one or more embodiments, the observable to be measured is the Pauli operator X, b) further includes b0) before b1) Pauli-Z ( ) gates are applied to the cat qubit, and b3) includes Pauli-X ( A gate is applied to the cat qubit.

[0017] According to one or more embodiments, the observable to be measured is a parity check operator, which is mapped to the Pauli operator X throughout a), and b) further includes b0) before b1) Pauli-Z ( ) gates are applied to the cat qubit, and b3) includes Pauli-X ( A gate is applied to the cat qubit.

[0018] According to one or more embodiments, the command circuit includes one or more microwave sources for performing a), by performing Pauli-X (…) by shifting the phase of one or more microwave sources by an angle π. A gate is applied to the cat qubit.

[0019] According to one or more embodiments, the observable to be measured is the Pauli operator X, and b3) includes Pauli-Y ( A gate is applied to the cat qubit.

[0020] According to one or more embodiments, the observable to be measured is a parity check operator, which is mapped to the Pauli operator X throughout a), and b3) includes transferring Pauli-Y ( A gate is applied to the cat qubit.

[0021] According to one or more embodiments, a first mode is constructed by driving a first mode by transmitting microwave radiation with a frequency equal to a first resonant frequency to at least one resonant portion via a command circuit. Hamiltonian operator H Z To execute Pauli-Y ( ) gate applied to or Pauli-Z ( A gate is applied to the cat qubit, where The amplitude and phase are generated by the first mode.

[0022] According to one or more embodiments, c) includes c1) using the shift operator The frequency applied to the cat qubit, and then c2) is delivered by the command circuit at a frequency equal to the resonant frequency. microwave radiation, resonant frequency Essentially equal to the second resonant frequency, used to parameterize the four-wave mixing nonlinear element driving the quantum system, thus constructing the four-wave mixing nonlinear element as follows: The Hamiltonian H, where It is generated by the amplitude of microwave radiation, and It is the annihilation operator of the second mode, the Hamiltonian H is generated by the vertical coupling between the first and second modes, and c3) performs heterodyne or zero-difference detection on the second mode.

[0023] According to one or more embodiments, c) includes c1) using the shift operator Applied to the cat qubit, c2) is then constructed as follows: The Hamiltonian H, where It is the frequency shift of each photon, and It is the annihilation operator of the second mode, the Hamiltonian H is generated by the cross Kerr term between the first and second modes, which is easy to modify the second resonant frequency, and then c3) measures the second resonant frequency.

[0024] According to one or more embodiments, c) includes c1) if the three-wave mixing nonlinear element or the four-wave mixing nonlinear element is a three-wave mixing nonlinear element, then a command circuit delivers microwave radiation equal to the absolute difference between the first resonant frequency and the second resonant frequency; or if the three-wave mixing nonlinear element or the four-wave mixing nonlinear element is a four-wave mixing nonlinear element, then a command circuit delivers microwave radiation equal to half the absolute difference between the first resonant frequency and the second resonant frequency, for parameterizing the driving of the three-wave mixing nonlinear element or the four-wave mixing nonlinear element, thereby constructing the three-wave mixing nonlinear element or the four-wave mixing nonlinear element as represented by The Hamiltonian H, where It is generated by the amplitude of microwave radiation, and It is the annihilation operator for the second mode, the Hamiltonian H is generated by the single-photon conversion between the first and second modes, and c2) performs heterodyne or null difference detection on the second mode.

[0025] According to one or more embodiments, c) includes c1) delivering microwave radiation with a frequency equal to the second resonant frequency to at least one resonant portion via a command circuit to drive the second mode, thereby constructing a three-wave or four-wave mixing nonlinear element as shown in the figure. jump operator c2) By commanding a circuit, microwave radiation with a frequency equal to the first resonant frequency is delivered to at least one resonant component to drive the first mode, thereby constructing the three-wave or four-wave mixing nonlinear element as follows: The Hamiltonian H, where The amplitude and phase are generated by the first mode, and the heterodyne or zero-difference detection is performed on the second mode by c3).

[0026] According to one or more embodiments, the transmission line is weakly coupled to a first mode (a), and (c) includes performing heterodyne or null detection to detect electromagnetic fields in the transmission line.

[0027] According to one or more embodiments, the three-wave or four-wave mixing nonlinear element is a four-wave mixing nonlinear element formed by an asymmetric tunneling superconducting quantum interference device.

[0028] According to one or more embodiments, a three-wave or four-wave mixing nonlinear element is a three-wave mixing nonlinear element formed by at least one loop including a first Josephson junction, a central inductor element, and a second Josephson junction, wherein the nonlinear superconducting quantum circuit is configured such that when a predetermined current of constant strength is applied, the second resonant frequency is substantially equal to twice the first resonant frequency. Attached Figure Description

[0029] Referring to the accompanying drawings, other features and advantages of the invention will become apparent from the following description, which is provided for illustrative and non-limiting purposes, wherein:

[0030] Figure 1 A quantum system according to the invention, arranged to measure observable quantities of cat qubits, is schematically illustrated.

[0031] Figure 2 It shows Figure 1 An example of a quantum system in which parameter dissipation stabilization is used to stabilize the cat qubit.

[0032] Figure 3 It shows Figure 2 The local electro-equivalence diagram of an example implementation of a nonlinear superconducting quantum circuit for a quantum system is shown, with only the ATS and two resonant sections shown.

[0033] Figure 4 It shows Figure 2 Another example of a nonlinear superconducting quantum circuit realization of a quantum system is shown in the local electro-equivalence diagram, which only shows the ATS and two resonant sections.

[0034] Figure 5 Partially shown Figure 2 An example implementation of a quantum system, showing a portion of the external environment,

[0035] Figure 6 Partially shown Figure 2 Another example implementation of a quantum system, in which a portion of the external environment is shown.

[0036] Figure 7 Partially shown Figure 1 Another embodiment of the quantum system in which resonant dissipation stabilization is used to stabilize the cat qubit.

[0037] Figure 8 Partially shown Figure 7 Example implementations of quantum systems,

[0038] Figure 9 A method for measuring observable quantities of cat qubits according to the present invention is shown.

[0039] Figure 10 It shows Figure 9 The mapping operation of the method,

[0040] Figure 11 This illustrates the measurement of the parity operator according to the first embodiment. Figure 10 The mapping operation is projected onto the cat coding space.

[0041] Figure 12 It shows Figure 11 The time evolution of different parameters involved in the mapping operation,

[0042] Figure 13 It shows the use of according to Figure 11 and 12 The experimental measurement of parity was performed using Wigner tomography.

[0043] Figure 14 It shows the method for execution Figure 13 Wigner tomography Figure 5 The physical chip layout for realizing quantum systems.

[0044] Figure 15 This illustrates the measurement of the parity operator according to the second embodiment. Figure 10 The mapping operation is projected onto the cat coding space.

[0045] Figure 16 It shows Figure 15 The time evolution of different parameters involved in the mapping operation,

[0046] Figure 17 This is shown in the context of measurement using the Pauli operator Y. Figure 10 The mapping operation is projected onto the cat coding space, and

[0047] Figure 18 The vertical coupling process of reading the value of the observable to be measured and mapped to the Pauli operator Z is shown.

[0048] The accompanying drawings and the following description include most of the features used for their positive and clearly defined purpose. Therefore, they not only aid in understanding the invention, but can also be used to aid in its definition if desired. Specific Implementation

[0049] Figure 1 A schematic diagram of quantum system 1 is shown.

[0050] Quantum system 1 is arranged to stabilize the cat qubit and measure its observables.

[0051] To date, the applicant's work has generally involved the stabilization of cat qubits. As previously mentioned, R. Lescanne et al. (2020) showed that such cat qubits can be stabilized by a nonlinear transition between two photons in a first mode a-storage mode or cat qubit mode and one photon in a second mode b-buffer mode.

[0052] Cat qubits are defined as being composed of so-called cat states. Across a two-dimensional manifold, where the cat state There are two coherent states. and Superposition: in:

[0053] cat-like The definition can be extended through parsing continuation. : in: and These are Fock states, which have 0 and 1 photons respectively.

[0054] Therefore, when At that time, the cat qubit manifold spans and All superpositions of Fock states.

[0055] It is known that stable cat qubits benefit from high noise bias, meaning that the probability of a bit flip is exponentially smaller than the probability of a phase flip. More precisely, the number of effective error channels (e.g., bit errors or "bit flips") is exponentially related to the "size" of the Schrödinger cat state of the cat qubit (i.e., the average number of photons). Together, they are suppressed. As mentioned earlier, this exponential suppression of bit-flipping errors comes at the cost of a linear increase in phase-flipping errors.

[0056] Based on current knowledge, this suppression should be applied to a large class of physical noise processes that have a local effect on the phase space of the harmonic oscillator. This includes, but is not limited to, photon loss, thermal excitation, photon dephasing, and various nonlinearities caused by coupling to the Josephson junction.

[0057] Recent experiments in the context of quantum superconducting circuits have observed this bit-flipping error with exponential suppression of the average photon number in cat states.

[0058] It is generally accepted that a single repeating code is sufficient to correct the remaining errors because bit-flip errors are rare enough; and more specifically, a phase-flip error-correcting code is sufficient to correct the remaining phase flips. This could be, for example, a repeating code defined in dibases or any other state-of-the-art error-correcting code.

[0059] Cat qubits can be stabilized or confined using the following exemplary schemes:

[0060] a) Parameter dissipation stabilization with jump operator ,in It is the two-photon dissipation rate. It is the photon annihilation operator, and It is a complex number that defines a cat qubit. The jump operator can be used to select a qubit with a dissipation rate. The lossy buffered mode b and a four-wave mixer (typically a Josephson junction or ATS) are coupled to the cat qubit mode and the Hamiltonian is constructed. To achieve this, in which It is the photon annihilation operator in buffered mode b, and It is the two-photon coupling rate, in In this case, by applying a frequency to the four-wave mixer pump and frequency The driver uses a buffered mode.

[0061] b) Kerhammill ,in It is the amplitude of the Kerhammeter. It is the photon annihilation operator, and It is the average number of photons.

[0062] c) Detuned Kelhamnton ,in The magnitude of the Kerhammillennium is 'a', and 'a' is the photon annihilation operator. It is the complex number that defines a cat qubit, and It is a detuning factor.

[0063] d) Two-photon exchange (TPE) Hamiltonian ,in It is the complex two-photon coupling rate. It is the photon annihilation operator. It is the complex number that defines a cat qubit, and It is the decrement and elevation operator for a two-level system. The Hamiltonian can be constructed in the same way as the parameter dissipation stabilization a).

[0064] e) Dissipative compression stabilization with jump operator ,in It is to compress the two-photon dissipation rate. It is the photon annihilation operator. It is the complex number that defines a cat qubit. and Complex compression parameters The modulus and the independent variable. The jump operator can be used by taking a variable with a dissipation rate. The lossy buffered mode b and a four-wave mixer (typically a Josephson junction or ATS) are coupled to the cat qubit mode and the Hamiltonian is constructed. To achieve this, in which It is the photon annihilation operator for mode b, and It is the compressed two-photon coupling rate, where, in Under these conditions, several pumps at frequency , and and frequency The driver uses a buffered mode.

[0065] f) Resonant dissipation stabilization with jump operator ,in It is the two-photon dissipation rate. It is the photon annihilation operator, and It is a complex number that defines a cat qubit. The jump operator can be used to select a qubit with a dissipation rate. The lossy buffer mode b and the three-wave mixer are coupled to the constructed Hamiltonian. The cat qubit mode a is used to achieve this, where It is the photon annihilation operator for buffered mode b, provided that the mode frequency is basically verified. and In frequency A driver in the lower buffer mode was added to it.

[0066] In the context of this invention, quantum system 1 is arranged to achieve parameter dissipation stabilization according to scheme a) or resonance dissipation stabilization according to scheme f). In both cases, the cat qubit is constructed as follows: Dissimilarity operators - Alternatively, use the jump operator to stabilize.

[0067] Parameter dissipation stabilization depends on generating the following terms:

[0068] The first term corresponds to the nonlinear transition between two photons (memory) in mode a and one photon (buffer) in mode b. This term requires a frequency of... The pump, of which It is the resonant frequency of the first mode a, and This is the resonant frequency of mode b in the second mode. Therefore, this pumping is two-photon pumping.

[0069] The second requirement is based on frequency. The second mode b is driven. Therefore, this drive is a two-photon drive.

[0070] Resonant dissipation stabilization corresponds to where In this specific case, and therefore the 2 to 1 photon conversion is not activated by parameterized pumping. Therefore, only frequency is required. The second mode b drives the construction of the jump operator. .

[0071] like Figure 1 As shown, quantum system 1 includes nonlinear superconducting quantum circuit 3 and command circuit 5.

[0072] The nonlinear superconducting quantum circuit 3 is arranged to perform possible three-wave mixing or four-wave mixing between a first mode a and a second mode b. In the following text, the first mode a serves as a memory carrying the cat qubit, while the second mode b serves as a buffer between the cat qubit and the external environment.

[0073] The first mode a and the second mode b each correspond to the inherent resonant frequencies of the nonlinear superconducting quantum circuit 3. Therefore, the first mode a and the second mode b each have their own resonant frequencies. The first mode a has a resonant frequency. Furthermore, the second mode b has a resonant frequency. ,in and These are the corresponding angular frequencies of the first mode a and the second mode b.

[0074] By referring to the first and second modes, it should be understood that the nonlinear superconducting quantum circuit 3 includes components operating in a superconducting state, which carry modes independently or in parallel with each other. In other words, the first mode a and the second mode b can be carried on different subsets of the components of the superconducting circuit or on the same subset of the components.

[0075] The nonlinear superconducting quantum circuit 3 is designed to withstand microwave radiation delivered by the command circuit 5 in order to construct various nonlinear interactions between the first mode a and the second mode b. The frequency of each microwave radiation is tuned to select a specific term within the rotating wave approximation.

[0076] The nonlinear superconducting quantum circuit 3 includes a nonlinear element 7 and at least one resonant part 9.

[0077] Nonlinear element 7 is arranged to construct a jump operator for stabilizing the cat qubit. As mentioned earlier, this stability can be achieved through parametric dissipation stabilization or resonant dissipation stabilization.

[0078] In achieving parameter dissipation stabilization, nonlinear element 7 is a four-wave mixing nonlinear element.

[0079] Four-wave mixing nonlinear elements are, for example, ATS. Those skilled in the art know that ATS can be used to construct 2-to-1 photon conversions, i.e., jump operators. The first item This is to perform parameter dissipation stabilization, as successfully demonstrated by R. Lescanne et al. (2020). More specifically, by frequency... The ATS is parameterized to achieve this 2-to-1 photon conversion between the first mode a and the second mode b. Advantageously, the pump frequency satisfies... To make the parameterized pump work as well as possible.

[0080] Compared to the first implementation of this stabilization scheme proposed by Z. Leghtas et al. (2015) and S. Touzard et al. (2018), in which the superconducting circuit element used as the four-wave mixer is a transmon with a single Josephson junction, the solution developed by R. Lescanne et al. (2020) utilizes an ATS design with a much lower cross-Kerr term than the transmon, and thus allows for the observation of exponential suppression of bit flipping.

[0081] Typically, an ATS has a flux line through which radiation can be delivered to modulate common flux and / or differential flux.

[0082] In achieving resonance dissipation stabilization, nonlinear element 7 is a three-wave mixing nonlinear element.

[0083] As mentioned above, the resonance condition corresponds to where And may be particularly advantageous in specific cases, such as those developed in the applicant's European patent application EP 21306965.1. In such resonant cases, the 2-to-1 photon conversion cannot be activated by parameterized pumping of a four-wave mixing nonlinear element; a three-wave mixing nonlinear element should be used instead.

[0084] However, as described below, measuring the observables of a cat qubit may involve constructing a Hamiltonian that requires four-wave mixing. Typically, longitudinal coupling between the first mode a and the second mode b can be used to perform the measurement of the observables of the cat qubit. However, the longitudinal term of the corresponding Hamiltonian... Four-wave mixing is still required for construction. Therefore, the resonant condition may require a three-wave mixing nonlinear element for stabilizing the cat qubit and a four-wave mixing nonlinear element for measuring the observable quantities of the cat qubit. Three-wave mixing can be achieved by... The flux operating point can be operated at different flux operating points or generated by adding another nonlinear element, as described in the aforementioned European patent application EP21306965.1.

[0085] The resonant section 9 is arranged to be connected to the nonlinear element 7 to provide the nonlinear superconducting quantum circuit 3 with a corresponding resonant frequency. and The first mode a and the second mode b. More specifically, the first mode a and the second mode b "participate" in the nonlinear element 7, meaning that some or all of the mode magnetic energy is stored in the nonlinear element 7. This participation can be quantified by the zero-point fluctuation of the superconducting phase at both ends of the ATS, for the first mode a labeled as For the second mode b, it is labeled as .

[0086] exist Figure 1 In the schematic diagram of quantum system 1 shown, the nonlinear superconducting quantum system 3 includes only one resonant part, namely resonant part 9. However, it should be understood here that the nonlinear superconducting quantum system 3 includes at least one resonant part, and typically includes two resonant parts to ultimately form two electromagnetic modes.

[0087] Command circuit 5 is configured to deliver microwave radiation.

[0088] In the context of quantum system 1, command circuit 5 is arranged such that its frequency is equal to the second resonant frequency. The radiation is delivered to the resonant part 9 to drive the second mode b, because both parameter dissipation stabilization and resonant dissipation stabilization require this driving of the second mode b.

[0089] Figure 2 The detailed architecture of quantum system 1 for achieving parameter dissipation stabilization is shown.

[0090] Therefore, the nonlinear element 7 is a four-wave mixing nonlinear element, and more precisely, an ATS, which is connected to the resonant part 9 of the linear microwave network b / a.

[0091] Those skilled in the art know that when at its flux operating point ( When biased (or vice versa), the Hamiltonian of ATS 7 has the following "sin-sin" form: in It is the total superconducting phase difference across the two ends of ATS 7. It is the zero-point oscillation of the phase of the first mode a at both ends of the ATS 7, and It is the zero-point oscillation of the phase of the second mode b at both ends of the ATS 7. It is the Josephson energy of the side junction, and It is the induced energy of the central inductor. This corresponds to the common flux modulation of the two loops of ATS 7, while This corresponds to differential flux modulation of the two loops of ATS 7. The former can be achieved by delivering microwave radiation out of phase via the flux lines of ATS 7, while the latter can be achieved by delivering microwave radiation in phase via the two flux lines of ATS 7.

[0092] Parametric pumping of the ATS 7 is typically accomplished via a pump common flux, since pump differential flux simply replaces the modes coupled to the ATS 7. As an example, this is achieved by frequency-based... , The nonlinear resonant component of the pump common flux and Hamiltonian is written into the rotating frame: This is typically the two-to-one photon exchange Hamiltonian required to construct two-photon stabilization.

[0093] Refer again Figure 2 The linear microwave network b / a enables the nonlinear superconducting quantum circuit 3 to participate in the corresponding resonant frequency in the ATS 7 when coupled via the linear coupler 11 to the ATS 7, which acts as an inductor. and The first mode a and the second mode b.

[0094] When the external DC magnetic field is set to enable Magnetic flux passes through a loop and When the magnetic flux passes through another loop, ensure that the ATS Hamiltonian has its "sin-sin" form. For clarity, the setting of applying an external DC magnetic field is not shown. Figure 2 Instead of being drawn on top, it can be applied via the two bottom mutual inductors of the ATS 7. A typical implementation includes a bias tee, which is interleaved with a DC current source to input DC current into the system while allowing microwave radiation to pass through.

[0095] Microwave source 13 is configured to modulate the common flux in ATS 7.

[0096] For this purpose, microwave network 15 is used to separate the radiation emitted by microwave source 13 and apply it to each node of ATS 7 with the correct phase. Alternatively, two different microwave sources can be used, each simply coupled to a single node of ATS 7, and their relative phase and amplitude are set to achieve the desired flux modulation.

[0097] When microwave source 13 is set to frequency At this time, the nonlinear superconducting quantum circuit 3 performs a 2-to-1 photon conversion between the first mode a and the second mode b. To convert this 2-to-1 photon conversion into two-photon dissipation, the second mode b is configured via a linear coupler 21 to have a frequency... The microwave filter 23 of the bandpass filter is selectively coupled to the load 19.

[0098] Alternatively, microwave filter 23 can be configured to frequency The band-stop filter can be placed in the external environment on one hand, and between the first mode a and the second mode b on the other hand, to isolate the first mode a, thereby preventing the first mode a from suffering additional losses from unwanted coupling with the load 19.

[0099] Alternatively, if (correspondingly) It can be configured as a low-pass (and correspondingly high-pass) filter. In other embodiments, microwave filter 23 can be omitted when coupling is established between load 19 and essentially only the second mode b. Therefore, those skilled in the art will understand that the first mode a has a high quality factor, while the second mode b has a low quality factor.

[0100] As mentioned earlier, the second mode b uses its resonant frequency Driven. This two-photon drive is set at a frequency Microwave source 17 is executed.

[0101] In the above text, load 19 can be considered as Figure 2 It is part of the command circuit 5, while the linear coupler 21 and microwave filter 23 can be regarded as part of the nonlinear superconducting quantum circuit 3.

[0102] In addition to stabilizing the cat qubit, command circuit 5 also enables the measurement of observable quantities of the latter. This is detailed below, especially in reference to... Figure 9 As described below, the measurement method, which is the subject of this invention, includes the mapping of the observable quantity to be measured to the Pauli operator Z and the measurement of the latter. For this purpose, as... Figure 2 As shown, the command circuit 5 also includes microwave source 25, microwave source 27 and microwave source 29.

[0103] Microwave source 25 is arranged to transmit frequency Microwave radiation is delivered to the linear microwave network b / a to drive the first mode a. This driving of the first mode a enables the construction of the nonlinear superconducting quantum circuit 3, represented as follows: Hamiltonian H Z , where complex rate The amplitude and phase are generated by the drive of the first mode a.

[0104] Such Hamiltonian H Z This can involve mapping the observable quantity to be measured. When constructed in conjunction with two-photon dissipation, the Hamiltonian H... ZPauli gates can be applied to the cat qubit depending on the size of the cat qubit and the driving phase of the first mode a, and more precisely, Pauli-Y gates or Pauli-Z gates can be applied to the cat qubit.

[0105] Furthermore, such Hamiltonian H Z This is how it can be used in measurements. When constructed in conjunction with two-photon dissipation, the Hamiltonian H... Z The value of the Pauli operator Z can be measured by performing heterodyne or zero-difference detection on the second mode b.

[0106] In addition to microwave source 13, microwave source 27 is also provided to modulate the common flux in ATS 7. As detailed below, such microwave source 27 can participate in the measurement in this way. Microwave source 27 is arranged to deliver microwave radiation so that nonlinear superconducting quantum circuit 3 constructs a Hamiltonian. For example, this Hamiltonian is generated via longitudinal coupling between a first mode a and a second mode b.

[0107] Finally, microwave source 29 is configured to modulate the differential flux in ATS 7 via microwave network 15. Microwave source 29 can be used to reduce the parasitic Hamiltonian term caused by microwave source 27. It should be noted that microwave network 15 is used for convenience but can be omitted, and microwave sources 29 and 27 can be directly applied to the two nodes of ATS 7, their relative phase and amplitude set to achieve the desired flux modulation.

[0108] For completeness, it can also be noted that microwave source 29 can be used instead of microwave source 17 to set the frequency. Microwave radiation is delivered to the linear microwave network b / a to drive the second mode b.

[0109] Figure 3 and Figure 4 Electrical equivalent diagrams of a corresponding embodiment of the nonlinear superconducting quantum circuit 3 in the form of a current circuit are shown. These electrical equivalent diagrams are partial, as they only represent the ATS 7 and the linear microwave network b / a, and therefore do not represent either the linear coupler 21 or the microwave filter 23. More specifically, the linear microwave network b / a includes a first resonant section 31 and a second resonant section 33.

[0110] The ATS 7 is implemented as is known in the art, for example in R. Lescanne et al. (2020). The ATS 7 includes a first Josephson junction 35 and a second Josephson junction 37 connected in parallel, and an inductor element 39 connected in parallel between them. As a result, the ATS 7 has two connection loops, each loop including a Josephson junction connected in parallel with the parallel inductor element. The inductor element 39 can be implemented geometrically or using junction chains. The ATS 7 biases the flux of its two loops in DC and AC. The DC bias sets the flux operating point of the ATS 7. It can operate near a so-called saddle point, which is an optimal point in frequency and has a small cross-Kerr term.

[0111] Both the first resonant section 31 and the second resonant section 33 are electrically coupled to the ATS 7. The first resonant section 31 will have a resonant frequency. The first mode a imparts nonlinear superconducting quantum circuit 3, while the second resonant part 33 will have a resonant frequency. The second mode b endows nonlinear superconducting quantum circuits with 3.

[0112] By "electrically coupled," it should be understood that there exists a short conductive portion connecting the first resonant portion 31 and the second resonant portion 33 to the ATS 7, i.e., a short conductive track or any other means that ensures a physically continuous conductive junction. The term "short" means that the conductive track has negligible impedance compared to the ATS 7 at the resonant frequency. and The first resonant portion 31 and the second resonant portion 33 are below. These short conductive portions correspond to... Figure 2 Linear coupler 11.

[0113] exist Figure 3 In one embodiment, the first resonant portion 31 includes a capacitor element 41 and an inductor element 43 connected in series. Similarly, the second resonant portion 33 includes a capacitor element 45 and an inductor element 47 connected in series.

[0114] exist Figure 4 In one embodiment, the first resonant portion 31 further includes a capacitor element 41 and an inductor element 43. However, in such an embodiment, the capacitor element 41 and the inductor element 43 are connected in parallel. Similarly, the capacitor element 45 and the inductor element 47 of the second resonant portion 33 are connected in parallel.

[0115] exist Figure 3 and Figure 4 In a corresponding embodiment, the microwave linear network b / a includes two resonant sections. However, as previously stated, the microwave linear network b / a may include only one resonant section arranged to generate both the first mode a and the second mode b.

[0116] Figure 5and Figure 6 Both showed Figure 2 Possible example implementations of quantum system 1. With Figure 3 and Figure 4 Conversely, the external environment is represented. ATS 7 serves as the central reference point, and the remaining components are connected to this central reference point. For this purpose, filtering and the external environment for mode b also exist. For simplicity, the microwave source is omitted, but it is related to... Figure 2 The circuits are arranged in the same way to operate.

[0117] exist Figure 5 In the circuit, the linear microwave network b / a is formed by a capacitor electrically coupled to ATS 7 to form the second mode b and a parallel LC resonator strongly coupled to ATS 7 to form the first mode a. The coupling with ATS 7 forms a linear coupler 11. A buffer is coupled to the external environment through capacitor 19.

[0118] exist Figure 6 In the diagram, the linear microwave network b / a consists of two series LC resonators electrically coupled to the ATS 7 to form a first mode a and a second mode b. A buffer is coupled to the external environment via capacitor 19. It can be noted that the diagrams of the ATS 7 and the linear microwave network b / a correspond to... Figure 3 As shown in the figure, the first mode a corresponds to the first resonant part 31, while the second mode b corresponds to the second resonant part 33.

[0119] The above discussion Figures 2 to 6 An embodiment of quantum system 1 is shown, wherein the nonlinear element 7 is an ATS, and wherein parametric dissipation stabilization is used to stabilize the cat qubit.

[0120] However, as mentioned above, the method for measuring observable quantities of the cat qubit according to the present invention can also be implemented using a three-wave mixing nonlinear element 7 and by stabilizing the cat qubit with resonant dissipation stabilization. Figure 7 and Figure 8 All of these involve such embodiments.

[0121] Figure 7 The quantum system 1 is partially shown. In this figure, a three-wave mixing nonlinear element 7 and at least one resonant section 9 are placed together to form a nonlinear superconducting quantum circuit 3. As previously described, this nonlinear superconducting quantum circuit 3 is arranged to inherently perform a 2-to-1 photon conversion between a first mode a and a second mode b, indicated here by a back-and-forth single arrow 49 and a double arrow 51.

[0122] Current source 53 is connected to nonlinear superconducting quantum circuit 3 via wires. This current source 53 can be considered as part of command circuit 5.

[0123] Current source 53 is directly connected to nonlinear superconducting quantum circuit 3, such that current applied by current source 53 flows through at least a subset of components of nonlinear superconducting quantum circuit 3. Current source 53 is configured to allow both three-wave mixing interaction and frequency matching conditions. .

[0124] Similar to Figure 2 The second mode b is coupled to the load 19 via a linear coupler 21, and this coupling makes the second mode b dissipative. The microwave source 17 is arranged to transmit a microwave at a frequency equal to... Microwave radiation is delivered to the resonant section 9 to drive the second mode b. As a reminder, both parameter dissipation stabilization and resonant dissipation stabilization require two-photon drive.

[0125] This also exists where it is configured to have a frequency. The microwave filter 23 is a bandpass filter. Alternatively, the microwave filter 23 can be configured to have a frequency... The band-stop filter can be placed in the external environment on the one hand, and between the first mode a and the second mode b on the other hand, to isolate the first mode a, thereby preventing the first mode a from suffering additional losses from unwanted coupling with the load 19.

[0126] Figure 8 It shows Figure 7 An example implementation of quantum system 1 is shown, and more specifically, a possible circuit for implementing a three-wave mixing nonlinear element is illustrated.

[0127] The three-wave mixing nonlinear element is formed by at least one loop 55 including a first Josephson junction 57, a center inductor 59, and a second Josephson junction 61.

[0128] When a predetermined current of constant intensity is applied by current source 53, the resonant frequency is... Basically equal to the resonant frequency Twice as much. Figure 8 The circuit shown is specially configured to symmetrically distinguish between the first mode a and the second mode b. The high symmetry of this circuit achieves improved quality in 2-to-1 photon conversion.

[0129] The central inductor 59 can be an inductor, a single Josephson junction, or an array of Josephson junctions. Therefore, the central inductor 59 can be arranged as a series loop between the first Josephson junction 57 and the second Josephson junction 61. The series arrangement may include a first internal node connecting one terminal of the first Josephson junction 57 to the terminal of the central inductor 59. The series arrangement may also include a second internal node connecting one terminal of the second Josephson junction 61 to the other terminal of the central inductor 59. The series arrangement may further include a closed node connecting the other terminal of the first Josephson junction 57 to the other terminal of the second Josephson junction 61.

[0130] At least one loop 55 may be connected to a common ground via a closed junction. The circuit may also include a first capacitor 63 and a second capacitor 65. The first capacitor 63 may be connected in parallel with a first Josephson junction 57 between the common ground and a first internal node of the loop. The second capacitor 65 may be connected in parallel with a second Josephson junction 61 between the common ground and a second internal node of the loop.

[0131] The first Josephson junction 57 and the second Josephson junction 61 are substantially identical, and the capacitor elements 63 and 65 are also substantially identical. Therefore, the symmetry of the circuit means that the first mode a is a symmetrical superposition of the two resonators (as indicated by the full arrow), and the second mode b is an anti-symmetrical superposition of the two resonators (as indicated by the dashed arrow). It can be noted that only the second mode b has the contribution of the transcenter inductor element 59, which is advantageously used to preferentially couple the external environment to the second mode b while isolating the first mode a from the external environment.

[0132] Now refer to Figure 9 Describe in detail the method for measuring observable quantities of cat qubits.

[0133] In quantum mechanics, an observable quantity is a physical quantity that can be measured or observed. Mathematically, an observable quantity is represented by a self-adjoint operator acting on the quantum state of an entity. The possible outcome of measuring an observable quantity is the eigenstate of the corresponding operator.

[0134] A typical example of observable quantities is: The position operator represents the position of a particle in space, and its eigenstates correspond to the possible positions of the particle. The momentum operator represents the momentum of a particle, and its eigenstates correspond to the possible moments of the particle. An energy operator represents the energy of a quantum entity, and its eigenstates correspond to the possible energy levels of the quantum entity. The spin operator, which represents the intrinsic angular momentum of a particle and takes three components. , and The vector operator is composed of the form; the eigenstates of the spin operator correspond to the possible values ​​of the particle's spin in different directions.

[0135] In the context of this invention, the observable quantity to be measured is the Pauli operator X, the Pauli operator Y, or the parity check operator of the cat qubit pattern in a cat qubit manifold.

[0136] Due to two-photon dissipation, the coherent state Stable, and only cat mode. Forming a standard orthogonal basis down to Therefore, the logical 0 and 1 states of the cat qubit base are defined as follows: when At that time, these states exponentially approach coherent states. Using this definition, the Pauli operator in a cat qubit manifold reads: Among them, state It is the following quantum superposition of coherent states:

[0137] Finally, the parity check operator takes the following form:

[0138] It can be noted that the projection of the parity check operator onto the cat qubit manifold is equal to the Pauli operator X. This invention utilizes this property to allow for Wigner tomography without transmission.

[0139] refer to Figure 9 In operation 900, quantum system 1 stabilizes the two-dimensional manifold carrying the cat qubits. If the cat qubit mode was initially in a state belonging to the cat qubit manifold, operation 900 leaves it unchanged. If the cat qubit mode was initially in a state outside the cat qubit manifold, the operation maps the parity check operator to the Pauli operator X of the stabilized manifold.

[0140] Therefore, command circuit 5 transmits microwave radiation to enable nonlinear superconducting quantum circuit 3 to construct a jump operator. .

[0141] Under the condition of parameter dissipation stabilization, the delivery frequency of microwave source 13 is equal to The microwave radiation parametrically drives the nonlinear element 7, and the microwave source 17 delivers at a frequency equal to... Microwave radiation is used to drive the second mode b.

[0142] Under the condition of resonant dissipation stabilization, the delivery frequency of microwave source 17 is equal to Microwave radiation is used to drive the second mode b, which is used to enable the nonlinear superconducting quantum circuit 3 to be controlled by the relation. The resulting intrinsic 2-to-1 photon conversion is converted into two-photon dissipation.

[0143] In operation 910, the observable to be measured is mapped to the Pauli operator Z.

[0144] By “mapping one observable to another”, it must be understood that quantum system 1 performs the process of transforming one observable into another.

[0145] From a mathematical perspective, if If it is the operator to be measured, then it is the unitary operator applied by quantum system 1. The process of description makes Therefore, the eigenvectors and eigenstates of the operator to be measured are mapped to the eigenvectors and eigenstates of the Pauli operator Z in the bijection. Thus, the measurement of the Pauli operator Z contains information about the operator to be measured. All the knowledge.

[0146] Figure 10 The mapping operation 910 implemented by quantum system 1 is shown in more detail, and the operations common to the various possible mappings that will be described in the remainder of the specification are also shown.

[0147] Figure 10 It also addresses a key aspect of the mapping implemented in the context of this invention: the variation caused by the size of the cat qubit manifold.

[0148] In the jump operator In the expression, It is the complex number that defines a cat qubit. More specifically, This corresponds to the size of the manifold that carries the cat qubit. Once stabilized, the cat qubit is carried in a manifold with an initial size of α. i In a two-dimensional manifold, and the jump operator reads... In the following text, Used as a parameter that can be changed by tuning the drive of the second mode b.

[0149] The mapping operation 910 specifically aims to map the observable quantity to be measured to a final value of α. TThe Pauli operator Z in the cat qubit manifold can be different from the initial size α of the cat qubit to be measured. i In order to improve measurement fidelity.

[0150] In fact, compared to Pauli operators X and Y, the advantage of Pauli operator Z is that its eigenstates are stabilized by two-photon dissipation. Therefore, their lifetime increases exponentially with the size of the cat qubit, allowing for measurement integration over very long periods and achieving a very good signal-to-noise ratio. If very long measurements are not possible, such as during quantum error correction periods, stabilizing the eigenstates of Pauli operator Z by two-photon dissipation allows for the α... T Increasing to arbitrarily large values ​​does not result in information loss. In this case, the eigenstates of the observable to be measured are mapped to coherent states. and coherent state and A large difference in photons This allows them to be effectively distinguished even within a short measurement time.

[0151] In operation 1100, quantum system 1 shrinks the cat qubit, thus reducing the size of the cat qubit manifold. .

[0152] To this end, quantum system 1 reduces or even shuts down the driving of the second mode b.

[0153] Command circuit 5 tunes the amplitude of microwave source 17 to make it so that It achieves a square modulus less than 1 and advantageously equal to 0. Since the cat qubit manifold is defined as... Therefore, operation 1100 will reduce the Pauli operator X from its initial size value α. i The cat qubit manifold is continuously mapped to the size The cat qubit manifold will have the Pauli operator Y starting from the initial size value α. i The cat qubit manifold is continuously mapped to the size The cat qubit manifold, and the Pauli operator Z is set from the initial size value α. i The cat qubit manifold is continuously mapped to the size Cat qubit manifold.

[0154] At the end of operation 1100, the cat qubit was essentially only in a state of... and Superposition of Fork states.

[0155] Basically, it should be understood that cat qubits are primarily... and A superposition of several Fock states. In fact, as mentioned above, the cat state... The definition can be extended to And in this case, the cat qubit manifold spans and All superpositions of Fock states. However, here, due to The size of the cat qubit manifold is almost zero, therefore some Fock states are... Or larger.

[0156] In operation 1200, quantum system 1 will - A Pauli gate is applied to the cat qubit.

[0157] Similar to the Pauli operators described above, each Pauli gate corresponds to a rotation about an axis around the Bloch sphere. The Pauli-X(θ) gate is a rotation about the x-axis, the Pauli-Y(θ) gate is a rotation about the y-axis, and the Pauli-Z(θ) gate is a rotation about the z-axis. To be executed by quantum system 1... - The Pauli gate depends on the observable to be measured. Strictly speaking, operation 1200 is the operation that maps the observable to be measured to the Pauli operator Z.

[0158] Finally, in operation 1300, quantum system 1 expands the cat qubit, thereby increasing the size of the cat qubit manifold. .

[0159] Therefore, quantum system 1 adds, or if applicable, enables the driving of the second mode b.

[0160] Command circuit 5 tunes the amplitude of microwave source 17 to make it so that Reaching the final size value Its square modulus is greater than or equal to 2: Jump operator reads

[0161] As explained below, satisfying final size value This indicates a significant improvement in measurement fidelity compared to existing technologies, due to the increase... The amplitude of the measurement signal is amplified, thus amplifying the measurement fidelity.

[0162] exist Figure 10 In the code, operations 1100, 1200, and 1300 are represented as sequential implementations. However, in practice, the parameters... The value is higher than all control setpoints of the microwave source (microwave source 17) arranged to drive mode b, so that the cat qubit reaches the setpoint value in steady state. The square modulus corresponds to the photon's brightness. However, in the transient state, it is related to the setpoint value. Compared to the square modulus, the actual average number of photons in a cat qubit Delay.

[0163] Therefore, in order to determine the application of operation 1200 -Optimal timing for Pauli-Men, considerations More relevant. As will be explained later, operation 1200... -Pauli should be in quantity It occurs when the value is less than 2.

[0164] Therefore, operation 1200 typically overlaps with both operations 1100 and 1300 in time.

[0165] In operation 920, quantum system 1 measures the value of the Pauli operator Z. As described above, such a Pauli operator Z has a size of α. T In the cat qubit manifold.

[0166] Due to the eigenstates-coherent states of the Pauli operator Z and - These are quasi-classical states that can have long lifetimes, a large number of photons, or both, and therefore can be read with high fidelity using a variety of techniques, including those developed below: i) Bitwise shift operators It is applied to the cat qubit. This is because the eigenstate of the observable to be measured is mapped to a coherent state. or Therefore, the bitwise shift operator Mapping the eigenstates of the observable quantity to be measured to coherent states. or Alternatively, the shift operator can be used. Applied to the cat qubit to map the eigenstate of the observable to be measured to a coherent state. or Next, command circuit 5 (more specifically, microwave source 27) delivers a frequency equal to the resonant frequency. The microwave radiation is constructed by parameterizing a four-wave mixing nonlinear element and represented as follows: The Hamiltonian H, where This is the amplitude of microwave source 27. Resonant frequency. It can be approximately equal to the resonant frequency of the second mode b. For example, the presence of longitudinal pumping (here, microwave source 25) can cause frequency... The so-called star-shaped offset. The Hamiltonian H is generated through longitudinal coupling between the first mode a and the second mode b. When the cat qubit is in a coherent state At that time, it will cause the second mode b to shift by a value. Finally, heterodyne or null difference detection is performed on the second mode b to measure the value of the Pauli operator Z. Clearly, this can be achieved by increasing the size of the cat qubit in operation 1300. To enable the detection signal Arbitrarily large, this thus acts as an amplification of the measurement. In the embodiment where parameter dissipation stabilization is achieved, the required four-wave mixing nonlinear element for achieving longitudinal coupling can be the nonlinear element 7, which is also used to stabilize the cat qubit. However, in the embodiment where resonant dissipation stabilization is achieved, the nonlinear element 7 is a three-wave mixing nonlinear element, and therefore, the four-wave mixing nonlinear element must be integrated into the quantum system 1 to achieve longitudinal coupling and thus perform the measurement operation. The four-wave mixing nonlinear element is as follows: Figure 2 In the case of ATS 7 shown, the radiation from microwave source 27 also causes parasitic drive of the buffer. The first-order expansion of the "sine-sine" Hamiltonian of the ATS does indeed give... To suppress this, radiation can be simultaneously applied by microwave source 29, making... . ii) More generally, with Any proportional Hamiltonian can be used when applying the shift operator. The value of the Pauli operator Z is then measured. One example is changing the flux operating point of the ATS so that its Hamiltonian causes a cross-Kerr term between mode a and mode b. ,in It is the frequency shift of each photon. As a result, when the cat qubit is in a coherent state... At that time, the frequency of the second mode b will change significantly. .again, This is used to amplify the measurement signal. Therefore, measuring the frequency of the second mode b will allow for efficient measurement of the observable quantity to be measured. Similarly, a cross-Kerr term is induced between the first mode a and the third mode c (not shown in the figure). Four-wave mixing will be suitable for this measurement. In particular, because... The provided amplification, the third mode c, can be very weakly coupled (i.e., with small...). The value is converted to the transmon of the first mode a, which reduces its effect on bit flipping, but a sufficiently large value is used. This makes the frequency shift of the transmon measurable. iii) If a four-wave mixing nonlinear element is used, the command circuit 5 (more specifically, microwave source 27) delivers a frequency equal to The microwave radiation, or if a three-wave mixing nonlinear element is used, then the command circuit 5 (more specifically, microwave source 27) delivers a frequency equal to... Microwave radiation is used to parameterize and drive nonlinear element 7, so that the nonlinear superconducting quantum circuit 3 is constructed as represented by The transformation Hamiltonian H, where It is the amplitude of microwave source 27, and It is the annihilation operator for mode b. The Hamiltonian H is generated through a single-photon conversion between mode a and mode b. Finally, heterodyne or null detection is performed on mode b to measure the value of the Pauli operator Z. iv) To further increase the signal-to-noise ratio, the converted Hamiltonian can be turned on towards the third mode c coupled to a three- or four-wave mixing nonlinear element (not shown in the figure). Simultaneously, two-photon drive generated by microwave source 17 is activated, and if necessary, two-photon pumping generated by microwave source 13 is activated to construct a coherent state over a very long period of time. and Stabilized jump operators This allows the measurement to be integrated over the same amount of time. v) Two-photon drive generated by microwave source 17, and, if necessary, two-photon pump generated by microwave source 13, is activated to construct a jump operator. And microwave source 25 directly delivers frequency Microwave radiation drives the first mode a, causing the Hamiltonian H to... Z Represented as -or ,in The amplitude and phase are generated by the drive of the first mode a. Finally, heterodyne or null detection is performed on the second mode b to measure the value of the Pauli operator Z. vi) The transmission line is weakly coupled to mode a, as done by Berdou et al. (2022), and the electromagnetic field in the transmission line is detected by null or heterodyne detection. To further increase the signal-to-noise ratio, two-photon drive generated by microwave source 17 can be enabled, and if necessary, two-photon pump generated by microwave source 13 can be enabled to construct a jump operator. The jump operator Achieving coherence over a very long period of time and Stabilization allows measurements to be integrated over the same amount of time.

[0167] The mapping from observables in the Pauli operators X, Y, and parity check to the Pauli operator Z has the advantage of forming a bijection. Therefore, the value of the Pauli operator Z gives the value of the observable to be measured.

[0168] Figure 11An example implementation of mapping operation 910 in an embodiment where the observable to be measured is a parity check operator is shown. The figure illustrates the rotation of the cat qubit in the Bloch sphere within the cat coding space and the change in the size of the cat qubit manifold.

[0169] As mentioned above, it should be noted that the jump operator is implemented throughout operation 900 by constructing it. To stabilize the cat qubit, the parity check operator is strictly mapped to the Pauli operator X. Strictly speaking, therefore... Figure 11 The mapping from the Pauli operator X to the Pauli operator Z is shown. However, it can be considered that... Figure 11 The operation is shown in the context of the measurement of the parity operator and in the context of the measurement of the Pauli operator X 910.

[0170] exist Figure 11 In the example shown, the stabilized cat qubit has an initial size .

[0171] In operation 1000, quantum system 1 will use Pauli-Z ( A gate is applied to the cat qubit. Therefore, the Pauli operator X is mapped to the Pauli operator Y.

[0172] Therefore, the command circuit 5 (more specifically, microwave source 25) delivers a frequency equal to... The microwave radiation is used to drive the first mode a to construct a representation as follows. Hamiltonian.

[0173] In operation 1100, quantum system 1 disables the driving of mode b, making the size of the cat qubit equal to zero: As mentioned earlier, the driving force of the second mode b can also be simply reduced. In this case, The value is tuned to a value whose square modulus is less than 1.

[0174] In operation 1200, quantum system 1 will use Pauli-X ( A gate is applied to the cat qubit. Therefore, the Pauli operator Y is mapped to the Pauli operator Z.

[0175] have The Pauli-X(θ) gate in the cat qubit manifold is particularly straightforward because it is equivalent to a rotation of angle θ in phase space. The orientation of the cat qubit in phase space is given by a quantum harmonic oscillator. The relative phase difference between the absolute phase of the oscillation and the absolute phase of the microwave source: in It is the absolute phase of microwave source 13, and It is the absolute phase of microwave source 17. To achieve Pauli-X (… Therefore, shifting the phase of one of microwave sources 13 and 17 by an angle π is sufficient, which can be done almost instantaneously at the software level with extreme precision.

[0176] In operation 1300, quantum system 1 activates the second mode b, causing the cat qubit to reach its final size. .exist Figure 11 In the example shown, the final size is Of course, it's also possible to simply add a driver for the second mode b. In this case, The value was tuned to the final size value. Its square modulus is greater than or equal to 2.

[0177] In an ideal quantum system 1, the theoretical requirements for performing operations 1100, 1200, and 1300 with 100% fidelity are as follows: - Operation 1200 -Pauli should be able to strictly control the cat qubit by... and The size span of the Fock state is The manifolds are completed because they have the correct rotational symmetry in phase space. Operation 1100 should precede operation 1200, and the size of the cat qubit manifold should be adiabatically reduced to [value missing]. . - Operation 1300 should follow Operation 1200, and the size of the cat qubit manifold should be adiabatically increased to up to .

[0178] However, in practice, true adiabatic operation is impossible because it would take an infinite amount of time. Therefore, the optimal time for operating at 1100 and 1300 will be determined by a trade-off between the following requirements: Compared to executing them fast enough so that single-photon loss does not degrade the fidelity of the operation, and with Compared to performing them slowly enough, these operations can be considered adiabatic. It is important to note that, as Albert et al. (2016) stated in the article " Holonomic Quantum Control with Continuous Variable Systems The fate of the Fock state, as explained in "Complete Quantum Control of Systems with Continuous Variables" (Physical Review Letters 116, 140502). and It is mainly related to the insulation conditions. In other words, when You can quickly deflate or inflate a cat's tires, but when... At that time, and This must be done slowly. Similarly, operation 1200 cannot be strictly applied in practice at a size of [missing value]. The cat qubit in the manifold is used to perform this, because it would take an infinite amount of time to get there during operation 1100. Nevertheless, when applied... - During Pauli gates, if the probability of a Fock state having two or more photons is small, it will perform well. To achieve good performance in operation 1200 while minimizing the durations of operations 1100 and 1300, the applicant found that by using a setpoint where the square modulus is less than 1 at the end of operation 1100... And through To achieve the optimal trade-off, a Pauli gate with operation 1200 is executed when the value is strictly less than 2. Furthermore, since cat qubits can have various initial states, and during the mapping process... The exact value depends on the initial state, therefore the condition This can be understood as satisfying all the initial states of interest for the cat qubit.

[0179] Figure 12 The diagram shows a sequence of pulses over a period of time, corresponding to the stabilization of the cat qubit and the mapping from the parity check operator to the Pauli operator Z. More specifically, the cat qubit initially stabilizes at a value of [value missing]. In a manifold of size Z, the Pauli operator Z is... In the cat qubit manifold. 2 to 1 photon conversion (by (Representation) is always on during the stable and mapping periods. Square modulus The curve allows for the initial observation of Pauli-Z ( The stability and application of the gate were then determined, and the contraction and expansion of the cat qubit were observed. When the value is less than 2, Pauli-X ( A gate is applied to the cat qubit.

[0180] The applicant has already used Figure 5 quantum circuits based on Figure 11 The parity operator of the cat qubit was experimentally measured using pulse sequences. Measurements of the parity operator can be used to perform Wigner tomography, which allows for the complete characterization of the quantum states of quantum harmonic oscillators, such as mode a. More precisely, in complex amplitude... The quantum state of the first mode a at position Wigner function Equivalent to in quantum state Already displaced : The average value of the subsequent parity check operators. For the initial state , , and Preparation of mode a, Wigner tomography measurements from top to bottom in Figure 13 As shown, it matches the expected exact match.

[0181] For this measurement, the applicant used the above procedure (i) to read the size as... The Pauli operator Z in a cat qubit manifold. Figure 18 The diagram shows the pulse sequence used to achieve this, and the shift of mode b, which depends on the number of photons in the first mode a. The representation of.

[0182] Figure 14 It is used to implement execution Figure 13 Wigner tomography Figure 5 An image showing the physical layout of the chip's circuitry.

[0183] Figure 15 An example implementation of mapping operation 910 in another embodiment is shown, where the observable to be measured is a parity check operator. The figure illustrates the rotation of the cat qubit in the Bloch sphere within the cat coding space and the change in the size of the cat qubit manifold.

[0184] Similar to Figure 11 The construction jump operator is implemented throughout operation 900. The stability of the cat qubit maps the parity check operator back to the Pauli operator X. Again, Figure 15 This illustrates the mapping from the Pauli operator X to the Pauli operator Z. Therefore, Figure 15 Operation 910 is shown in the context of measurement of the parity operator and in the context of measurement of the Pauli operator X.

[0185] exist Figure 15 In the example shown, the stabilized cat qubit has an initial size .

[0186] In operation 1100, quantum system 1 disables the driving of mode b. The cat qubit reaches size [value missing]. As mentioned earlier, the driving force of the second mode b can also be simply reduced. In this case, The value is tuned to a value whose square modulus is less than 1.

[0187] In operation 1200, quantum system 1 will use Pauli-Y ( A gate is applied to the cat qubit. Therefore, the Pauli operator X is directly mapped to the Pauli operator Z. For this purpose, command circuit 5 (more specifically, microwave source 25) delivers a frequency equal to... The microwave radiation is used to drive the first mode a to construct a representation as follows. Hamiltonian H Z .

[0188] In operation 1300, quantum system 1 activates the second mode b, causing the cat qubit to reach its final size. .exist Figure 15 In the example shown, the final size is Of course, it's also possible to simply add a driver for the second mode b. In this case, The value is tuned to the final size value. Its square modulus is greater than or equal to 2.

[0189] Figure 16 The diagram shows a sequence of pulses over a period of time, corresponding to the stabilization of the cat qubit and the mapping from the parity check operator to the Pauli operator Z. More specifically, the cat qubit initially stabilizes at a value of [value missing]. In a manifold of size Z, the Pauli operator Z is... In the cat qubit manifold. 2 to 1 photon conversion (by (This indicates that) it is always on during the stable and mapping periods. Square modulus The curves allow for the initial observation of stability, followed by the observation of the contraction and expansion of the cat qubit. When the value is less than 2, Pauli-Y ( A gate is applied to the cat qubit.

[0190] at last, Figure 17 An example implementation of mapping operation 910 is shown in an embodiment where the observable to be measured is the Pauli operator Y. The figure illustrates the rotation of the cat qubit in the Bloch sphere within the cat encoding space and the change in the size of the cat qubit manifold.

[0191] It seems that the mapping operation is similar to Figure 11 The mapping operation shown does not require mapping the Pauli operator X to the Pauli operator Y, because the observable to be measured is the latter.

[0192] The method of this invention is not merely about mapping the observable to be measured to the Pauli operator Z. In fact, the operation that expands the cat qubit—which corresponds to the drive to increase or enable the second mode b—effectively makes it possible to measure arbitrarily large sizes (i.e., making the square modulus...). Values ​​greater than or equal to 2 The Pauli operator Z in a cat qubit manifold. Such a size allows for the amplification of large quantities in the signal-to-noise ratio, eventually until the measurement becomes a single measurement point, i.e., when the difference in the number of photons between the logical states corresponding to the corresponding possible values ​​of the observable to be measured is high enough to avoid any ambiguity. This is in Figure 18 As shown in the figure, where with From the corresponding As the histogram increases, the histograms of the longitudinally coupled measurements using process (i) become increasingly separated.

[0193] With reduced overlap, the signal-to-noise ratio increases, and high-fidelity single measurements may be possible.

[0194] This method is particularly different from the parity check measurement proposed in patent application US 2022 / 0156622 A1, which describes a simple reduction of the cat qubit to represent the cat state in paragraphs

[0220] -

[0226] . Mapped to and cat state Mapped to . and The Fock states differ by only one photon, which is too low to distinguish them, and all indications suggest that, as presented in this prior art document, measurements with longitudinal coupling cannot achieve a satisfactory signal-to-noise ratio. Conversely, measurements with even minimal boundaries... The measurement method (i) leads to The difference in individual photons, an order of magnitude larger, leads to Figure 18 The histogram of separation in the data.

[0195] Furthermore, the measurement method described in this invention has the advantage of selecting an appropriate size for the cat qubit manifold for each mapping embodiment, especially for - The application of Pauli gates. For example, the Pauli-Z gate is more reliably implemented in cat qubit manifolds with a square modulus greater than 1, while the Pauli-X and Pauli-Y gates can only be implemented in cat qubit manifolds with a square modulus less than 1.

Claims

1. A method for measuring observables of a cat qubit implemented by a quantum system (1), wherein the observables to be measured are among parity check operators, Pauli operators X and Y, The quantum system (1) includes a command circuit (5) and a nonlinear superconducting quantum circuit (3). The command circuit is used to deliver microwave radiation. The nonlinear superconducting quantum circuit (3) includes a three-wave or four-wave mixing nonlinear element (7) and at least one resonant section (9). The three-wave or four-wave mixing nonlinear element (7) is connected to the at least one resonant section (9). The nonlinear superconducting quantum circuit (3) has a first mode (a) with a first resonant frequency and a second mode (b) with a second resonant frequency. The method includes the following operations: a) The command circuit (5) delivers microwave radiation (900) at a frequency equal to the second resonant frequency to the at least one resonant part (9) to drive the second mode (b), thereby constructing the three-wave or four-wave mixing nonlinear element (7) as shown in the figure. jump operator This stabilizes the two-dimensional manifold carrying the cat qubits, wherein, It is the two-photon dissipation rate. It is the annihilation operator of the first pattern (a), and It is a complex number produced by the delivered microwave radiation. b) Mapping (910) the observable to be measured to the Pauli operator Z, the mapping (910) depending on the observable to be measured, and including at least: b1) turning off or reducing (1100) the drive of the second mode (b) to make When its square modulus reaches a value less than 1, and then b2) enable or increase (1300) the drive of the second mode (b) to make Its square modulus is greater than or equal to 2. The mapping (910) further includes b3) during b1) or b2) in When the time is less than 2 - A Pauli gate is applied (1200) to the cat qubit, and c) Measure the value of the Pauli operator Z as described in (920).

2. The method according to claim 1, wherein, The observable to be measured is the Pauli operator Y, and wherein b3) includes Pauli-X ( A gate (1200) is applied to the cat qubit.

3. The method according to claim 1, wherein, The observable to be measured is the Pauli operator X, where b) further includes b0) before b1) Pauli-Z( A gate (1000) is applied to the cat qubit, and wherein, b3) includes applying Pauli-X ( A gate (1200) is applied to the cat qubit.

4. The method according to claim 1, wherein, The observable to be measured is the parity check operator, which is mapped to the Pauli operator X throughout a), wherein b) further includes b0) before b1) Pauli-Z( A gate (1000) is applied to the cat qubit, and wherein, b3) includes applying Pauli-X ( A gate (1200) is applied to the cat qubit.

5. The method according to any one of claims 2 to 4, wherein the command circuit (5) comprises one or more microwave sources (13, 17) for performing a), wherein, Pauli-X ( The gate (1200) is applied to the cat qubit by shifting the phase of one or more microwave sources (13, 17) by an angle π.

6. The method according to claim 1, wherein, The observable to be measured is the Pauli operator X, and wherein b3) includes Pauli-Y ( A gate (1200) is applied to the cat qubit.

7. The method according to claim 1, wherein, The observable to be measured is the parity check operator, which is mapped to the Pauli operator X throughout a), and wherein b3) includes transferring Pauli-Y ( A gate (1200) is applied to the cat qubit.

8. The method according to any one of claims 3 to 7, wherein, Pauli-Y ( ) gate applied (1200) or Pauli-Z ( The application of gate (1000) to the cat qubit is performed by delivering microwave radiation with a frequency equal to the first resonant frequency to the at least one resonant part (9) by the command circuit (5) to drive the first mode (a), thereby constructing a mode represented as shown in Figure 1000. Hamiltonian H Z ,in, The amplitude and phase are generated by the first mode (a).

9. The method according to any one of the preceding claims, wherein, c) Including c1) The bitwise shift operator The frequency c2) is applied to the cat qubit, and then c2) is delivered by the command circuit (5) at a frequency equal to the resonant frequency. microwave radiation, the resonant frequency Essentially equal to the second resonant frequency, used to parameterize the four-wave mixing nonlinear element (7) driving the quantum system (1), thereby constructing the four-wave mixing nonlinear element (7) as follows: The Hamiltonian H, where, It is generated by the amplitude of the microwave radiation, and It is the annihilation operator of the second mode (b), the Hamiltonian H is generated by the vertical coupling between the first mode (a) and the second mode (b), and c3) performs heterodyne or zero-difference detection on the second mode (b).

10. The method according to any one of claims 1 to 8, wherein, c) Including c1) The bitwise shift operator Applied to the cat qubit, c2) is then constructed as follows: The Hamiltonian H, where, It is the frequency shift of each photon, and It is the annihilation operator of the second mode (b), the Hamiltonian H is generated by the cross Kerr term between the first mode (a) and the second mode (b), the cross Kerr term is easy to modify the second resonant frequency, and then c3) the second resonant frequency is measured.

11. The method according to any one of claims 1 to 8, wherein, c) Including c1) If the three-wave or four-wave mixing nonlinear element (7) is a three-wave mixing nonlinear element, then the command circuit (5) delivers microwave radiation equal to the absolute difference between the first resonant frequency and the second resonant frequency, or if the three-wave or four-wave mixing nonlinear element (7) is a four-wave mixing nonlinear element, then the command circuit (5) delivers microwave radiation equal to half the absolute difference between the first resonant frequency and the second resonant frequency, for parameterizing the driving of the three-wave or four-wave mixing nonlinear element (7), thereby constructing the three-wave or four-wave mixing nonlinear element (7) as represented by c1) The Hamiltonian H, where, It is generated by the amplitude of the microwave radiation, and c1) is the annihilation operator of the second mode (b), the Hamiltonian H is generated by single-photon conversion between the first mode (a) and the second mode (b), and c2) performs heterodyne or null detection on the second mode (b).

12. The method according to claims 1 to 8, wherein, c) Including c1) the command circuit (5) delivers microwave radiation with a frequency equal to the second resonant frequency to the at least one resonant part (9) to drive the second mode (b), thereby constructing the three-wave or four-wave mixing nonlinear element (7) as shown in c1) jump operator c2) The command circuit (5) delivers microwave radiation with a frequency equal to the first resonant frequency to the at least one resonant part (9) to drive the first mode (a), thereby constructing the three-wave or four-wave mixing nonlinear element (7) as shown in the figure. The Hamiltonian H, where, The amplitude and phase are generated by the first mode (a), and the heterodyne or zero-difference detection is performed on the second mode (b) by c3).

13. The method according to claims 1 to 8, wherein the transmission line is weakly coupled to the first mode (a), wherein, c) Includes performing heterodyne or zero-difference detection to detect electromagnetic fields in the transmission line.

14. The method according to any one of the preceding claims, wherein the three-wave or four-wave mixing nonlinear element (7) is a four-wave mixing nonlinear element formed by an asymmetric tunneling superconducting quantum interference device.

15. The method according to any one of the preceding claims, wherein the three-wave or four-wave mixing nonlinear element (7) is a three-wave mixing nonlinear element formed by at least one loop (55), the at least one loop (55) comprising a first Josephson junction (57), a central inductor element (59), and a second Josephson junction (61), and the nonlinear superconducting quantum circuit (3) is configured such that when a predetermined current of constant strength is applied, the second resonant frequency is substantially equal to twice the first resonant frequency.

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