Methods for determining the preparation of cat states and gottesman-kitav-preskill states

Abstract

This disclosure relates to the preparation of cat states and Gottesman-Kitaev-Preskill (GKP) states. Entangled states are created by feeding an initial input state into a first input of a beam splitter and an initial compressed vacuum state into a second input. The entangled state is then measured at a first output using a detector to cause it to collapse into an output state at the second output. Subsequent entangled states are then created by feeding the output state into the first input; updating the compression direction to prepare a newer compressed vacuum state; and feeding the updated compressed vacuum state into the second input. The subsequent entangled states are then measured to produce a subsequent output state for feeding into the first input; and cat states are prepared iteratively by repeating the steps of creating and measuring subsequent entangled states.

Description

Cross-references to related applications

[0001] This application claims priority to Australian Provisional Patent Application No. 2023903233, filed on 9 October 2023, the contents of which are incorporated herein by reference in their entirety. Technical Field

[0002] This disclosure relates to the preparation of cat states and Gottesman-Kitaev-Preskill (GKP) states. More specifically, but not exclusively, cat states and GKP states are optical (quantum) states. Background Technology

[0003] Quantum information devices, such as quantum computers, suffer from poor quantum information stability due to errors such as noise, and many error correction strategies have been proposed. One such strategy involves encoding quantum information for error correction. More specifically, some methods embed a finite-dimensional code space into an infinite-dimensional Hilbert space of the system described by continuous quantum variables. This embedding can be accomplished using quantum states such as cat states and Gottesman-Kitaev-Preskill (GKP) states. The properties of these quantum states and the embedding schemes enable the correction of quantum information in the event of errors.

[0004] One condition for these methods is the presence of a large number of photons in the code. That is, the state exhibits large amplitude or large compression. The uncertainties of position q and momentum p satisfy... "Compression" then refers to reducing something at the expense of another. or For large compressions, the probability of inherent errors caused by defects in approximate GKP codewords decreases exponentially. The degree of compression has a substantial impact on the performance of error-correcting codes. Therefore, a method is needed to generate superpositions of highly compressed quantum states (especially highly compressed cat states that can be used for quantum error correction and to create GKP states).

[0005] However, these quantum states, especially the large-amplitude and highly compressed cat states, are difficult to prepare. For example, schemes in quantum optics can only generate states with low success probabilities, small amplitudes, and low compression. Therefore, such useful quantum states cannot be properly generated, which limits the development of quantum information devices because quantum errors remain a problem. Thus, a method is needed that can deterministically generate these quantum states with desired properties, meaning a guaranteed generation of these states.

[0006] Any discussion of documents, actions, materials, equipment, articles, etc. already included in this specification should not be construed as an admission that any or all of these matters form part of the prior art or are common general knowledge in the field relating to this disclosure, as they existed prior to the priority date of each of the appended claims.

[0007] Throughout this specification, the word “comprise” or variations thereof such as “comprises” or “comprising” shall be understood to imply inclusion of the said element, integer or step, or group of elements, integers or steps, but not to exclude any other element, integer or step, or group of elements, integers or steps. Summary of the Invention

[0008] According to this disclosure, a method for preparing a cat-like state is provided, the method comprising: An entangled state is generated by feeding an initial input state into the first input of the beam splitter and an initial compressed vacuum state into the second input of the beam splitter, the compressed vacuum state being parameterized by the compression direction; The entangled state is measured at the first output of the beam splitter using a detector, so that the entangled state collapses into the output state at the second output of the beam splitter; The subsequent entangled state is created in the following way: The output state is fed into the first input; Update the compression direction to prepare a new compressed vacuum state; and The updated compressed vacuum state is fed into the second input; Measuring the subsequent entangled state to generate a subsequent output state for feeding into the first input; and The cat state is iteratively prepared at the second output by repeating the steps of creating and measuring subsequent entangled states.

[0009] A mode for generating and measuring entangled states by mixing the input state with a compressed vacuum state on a beam splitter with high transmittance is advantageous because this effectively performs weak phase measurements. Repeated weak phase measurements can be used to deterministically prepare cat states; therefore, cat states with large amplitudes can be prepared by feeding the output state to the first input and repeating effective phase measurements.

[0010] In some embodiments, the initial input state is a quantum state with dual phase rotation symmetry.

[0011] In some embodiments, the method further includes measuring the Fock number parity of the first state to prepare an initial input state with dual-phase rotational symmetry.

[0012] In some embodiments, the initial input state is a combination of multiple quantum states, each of which has dual rotational symmetry.

[0013] In some embodiments, the initial input state is a single-mode Fock state.

[0014] In some embodiments, the single-mode Fock state is a compressed single-mode Fock state.

[0015] In some embodiments, the method further includes adjusting the transmittance of the beam splitter based on the output state or subsequent output state during repetition.

[0016] In some embodiments, the compressed vacuum state is parameterized by the compression amplitude; and the method further includes updating the compression amplitude based on the result of measuring the entangled state during repetition or the result of measuring the subsequent entangled state.

[0017] In some embodiments, updating the compression direction includes randomly selecting a direction.

[0018] In some embodiments, updating the compression direction includes updating the compression direction to a direction different from each previously repeated compression direction.

[0019] In some embodiments, the compression direction is updated based on the result of measuring the entangled state during repetition or the result of measuring the subsequent entangled state.

[0020] In some embodiments, the transmittance of the beam splitter is between 0.9 and 1, such that the total transmittance is between 0.4 and 0.6.

[0021] In some embodiments, the steps of generating subsequent entangled states and measuring subsequent entangled states are repeated at least four times.

[0022] In some embodiments, the cat state and the initial input state are optical states.

[0023] In some embodiments, the method further includes increasing the amplitude or compression amplitude of the cat state by performing the following steps: An entangled cat state is generated by feeding a cat state into the entanglement device, along with another cat state or another compressed vacuum state prepared using any of the methods described in the previously described embodiments; and The entangled cat state is measured at one output of the entanglement device using another detector, causing the entangled cat state to collapse into an improved cat state at the other output of the entanglement device. The improved cat state has a larger amplitude or compression amplitude than the original cat state.

[0024] According to this disclosure, a method for quantum error correction is provided, the method comprising: For each of the multiple codewords, multiple codewords are generated in a continuous variable quantum space by applying one or more of rotation, compression, displacement, or parity measurement to a cat state prepared using any of the methods described in the previously described embodiments. Encoding quantum information into the plurality of codewords to generate encoded quantum information; and Errors in the encoded quantum information are corrected based on the encoded quantum information or the plurality of codewords.

[0025] According to this disclosure, a method for preparing the Gottesman Kitaev-Preskill (GKP) state is provided, the method comprising: The first cat state and the second cat state are prepared using any of the methods described in the previously described embodiments; Another entangled state is created by feeding the first cat state and the second cat state into another entanglement device; and The other entangled state is measured at one output of the other entangled device using another detector, so that the other entangled state collapses into the GKP state at another output of the other entangled device.

[0026] In some embodiments, the method further includes correcting the first and second cat states before creating another entangled state by rotating and compressing the first and second cat states.

[0027] In some embodiments, the method further includes correcting the GKP state by rotating and compressing the GKP state.

[0028] In some embodiments, the method further includes: Repeat the method of any of the previously described embodiments to prepare the second GKP state; and Breeding of the GKP state and the second GKP state to produce a higher GKP state.

[0029] According to this disclosure, a method for quantum error correction is provided, the method comprising: For each of the multiple codewords, multiple codewords are generated in a continuous variable quantum space by applying one or more of rotation, compression, displacement or parity measurement to a GKP state prepared using any of the methods described in the previously described embodiments. Encoding quantum information into the plurality of codewords to generate encoded quantum information; and Errors in the encoded quantum information are corrected based on the encoded quantum information or the plurality of codewords.

[0030] According to this disclosure, a system for preparing a cat-like state is provided, the system comprising: The initial input state source is configured to generate the initial input state; A compressed vacuum source is configured to generate an initial compressed vacuum state and an updated compressed vacuum state, wherein the initial compressed vacuum state and the updated compressed vacuum state are parameterized by the compression direction; A beam splitter includes a first input, a second input, a first output, and a second output, wherein the beam splitter is configured to create entangled states when an initial input state is fed into the first input and an initial compressed vacuum state is fed into the second input; and The detector is configured to measure the entangled state at a first output so that the entangled state collapses into the output state at a second output; The beam splitter is further configured to create a subsequent entangled state when feeding the output state to the first input and feeding an updated compressed vacuum state to the second input; the detector is further configured to measure the subsequent entangled state to generate a subsequent output state for feeding into the first input; and to iteratively prepare the cat state at the second output by repeating the steps of creating the subsequent entangled state and measuring the subsequent entangled state.

[0031] In some embodiments, the detector is a photon number-resolved detector.

[0032] According to this disclosure, a system for preparing GKP states is provided, the system comprising: The system of some previously described embodiments is configured to prepare a first cat state and a second cat state; Another entanglement device is configured to create another entangled state while feeding the first cat state and the second cat state; and Another detector is configured to measure the other entangled state at one output of the other entanglement device, such that the other entangled state collapses into the GKP state at another output of the entanglement device.

[0033] In some embodiments, another entanglement device is a CNOT logic gate or a beam splitter.

[0034] In some embodiments, another detector is a zero-difference detector, which is configured to perform zero-difference detection on orthogonal components perpendicular to the compression direction or amplitude direction of the first cat state or the second cat state. Attached Figure Description

[0035] The following figures will be used as a reference for the example description:

[0036] Figure 1 The Wigner quasi-probability distribution function of the single-mode optical state is shown.

[0037] Figure 2a A schematic circuit is shown for performing measurements using a compressed vacuum state as input.

[0038] Figure 2b A schematic circuit for performing phase measurements is shown.

[0039] Figure 2c It shows that for The evolution of the phase distribution of random input states using a forward circuit (to prepare a rotating compressed vacuum) or a reverse circuit (to measure a rotating compressed vacuum).

[0040] Figure 2d It shows that for The evolution of the phase distribution of random input states using a forward circuit (to prepare a rotating compressed vacuum) or a reverse circuit (to measure a rotating compressed vacuum).

[0041] Figure 3a A system for evolving a quantum state into a cat state is shown.

[0042] Figure 3b The method for preparing in one iteration is shown. Figure 3a The cat-like system in the middle.

[0043] Figure 3c This demonstrates the preparation method within multiple iterations. Figure 3a The cat-like system in the middle.

[0044] Figure 4 A method for preparing the cat state is shown.

[0045] Figure 5a This shows the input for the Fock state. Phase probability distribution The evolution of.

[0046] Figure 5b Showing from Figure 5a The Wigner function for the initial Fock state.

[0047] Figure 5c This demonstrates the use of Fock state input. The Wigner function for the cat state is prepared using the disclosed method.

[0048] Figure 6a The initial compressed Fock state with 6dB compression is shown. Phase probability distribution The evolution, in which n =3.

[0049] Figure 6b Showing from Figure 6a The Wigner function for the initial compressed Fock state.

[0050] Figure 6cThe initial compressed Fock state with 6dB compression is shown. The Wigner function for the cat state prepared using the disclosed method, wherein... n =3.

[0051] Figure 7a The phase probability distribution of any input state is shown. .

[0052] Figure 7b It shows Figure 7a The Wigner function can be used with any input state.

[0053] Figure 7c It shows the use of Figure 7a and Figure 7b The Wigner function is used for arbitrary input states and cat states prepared using the disclosed method.

[0054] Figure 8a This shows the relationship between the success probability and the Fock number cutoff, which indicates the probability of success for fixed compression and n The smallest possible amplitude of the kitten that can be obtained for a given success probability.

[0055] Figure 8b It shows that for a given n Compressed Fock State as a function of Fock State Compression (dB) The average number of photons.

[0056] Figure 9a This shows the effect of a fixed number of Fock states. As the total transmittance The average fidelity of the function in 50 experiments of the disclosed method.

[0057] Figure 9b It shows that for Figure 9a The same parameter in the total transmittance The compression of the function in dB (numerically derived from...) (variance in the data).

[0058] Figure 10 A system for preparing the Gottesman-Kitaev-Preskill (GKP) state is shown.

[0059] Figure 11 A method for preparing GKP states is shown.

[0060] Figure 12 It shows Figure 10 The preferred embodiment of the system shown is illustrated.

[0061] Figure 13Numerical results for GKP breeding from cat forms prepared using the methods disclosed herein are shown.

[0062] Figure 14 The initial photon number is shown. n The (average) optimal entanglement fidelity of the function, assuming transmittance at the end of breeding. The pure loss error channel and the best recovery operation.

[0063] Figure 15 The fidelity before each detector for different loss amounts is shown in relation to the input Fock state. n The relationship.

[0064] Figure 16a A graph of the Wigner function is shown, illustrating an ideal input state consisting of cat states (i.e., a pure state with exact double rotational symmetry) and a typical output state for the ideal input state.

[0065] Figure 16b The graph shows the Wigner function for non-ideal input states (i.e., pure states without exact double rotational symmetry) and the corresponding typical outputs for non-ideal inputs.

[0066] Figure 17 The average fidelity of the prepared compressed cat state is shown. With the number of iterations The relationship. Detailed Implementation

[0067] Quantum technologies promise significant advantages across numerous fields, such as exponential speedups in computation time and absolute security in communication. However, the pervasive noise in quantum systems must be overcome before these technologies can be widely adopted. In the context of quantum computing, quantum communication, or other tasks in quantum information processing, quantum error correction offers a possible solution to address noise in quantum processors.

[0068] Encoding of quantum information can be characterized as discrete variables (DV) or continuous variables (CV). DV encoding is limiting because it requires many DV systems to encode reliable qubits or quantum d-bits, such as many auxiliary qubits, to create a logical qubit. In contrast, CV encoding cleverly utilizes the infinite-dimensional Hilbert space of resonators (such as photons) and therefore requires fewer moving parts. For example, they can encode quantum information as CV orthogonal components, such as position orthogonal components. Orthogonal components of momentum It has a continuous spectrum. In principle, the CV protocol provides easier manipulation of quantum states and is compatible with existing optical telecommunications infrastructure for quantum communication.

[0069] A subset of CV states is Gaussian states, which are states that can be represented using a Gaussian function. More specifically, a Gaussian state is a state whose Wigner quasi-probability distribution function is a (multivariate) Gaussian function (i.e., of the form...). Those states that are functions of Gaussian CV states. Examples of Gaussian CV states are coherent states, squeezed states, and thermal states.

[0070] Non-Gaussian states lead to more exotic states. These can be composed of carefully chosen subsets of the fully infinite-dimensional Hilbert space of CV modes to encode DV systems, such as cat states and Gottesman-Kitaev-Preskill (GKP) states. Cat states are superpositions of coherent states with opposite phases, while GKP states are multi-component superpositions of many compressed coherent states. These states possess error-correcting properties, which can be used to prevent noise in quantum devices such as quantum computers or quantum repeater nodes. Cat codes (error correction schemes utilizing cat states) prevent discrete pure loss errors and dephase errors, while GKP codes (error correction schemes utilizing GKP states) effectively prevent random shift errors in phase space as well as loss. Using cat states, quantum error correction can be realized in optical mechanisms. However, large-amplitude cat states and high-quality GKP states are difficult to prepare, especially in optics, particularly when attempting to produce these states deterministically rather than probabilistically.

[0071] This disclosure provides a method and system for deterministically preparing cat states. More specifically, an iterative scheme is used to deterministically generate large-amplitude cat states by using only Gaussian operations and photon number measurements from sources with even / odd photon number parity states. Using only Gaussian operations and photon number measurements greatly simplifies the preparation of these states, as these operations and measurements can be performed using readily available apparatus and devices. Furthermore, the cat states prepared using the method and system described herein can have approximately the same amplitude as the initial Fock state, meaning that as long as the initial input state is large, the prepared cat states will have large amplitudes.

[0072] Phase measurements can be used to deterministically generate cat states. However, such phase measurements are unavailable, especially in optical cases. The disclosed method overcomes this by providing a measurement similar to a phase measurement, thereby enabling the deterministic preparation of cat states. In particular, the disclosed method can be shown as selecting intervals on the phase distribution. The two peaks result in two cat states.

[0073] Importantly, the disclosed method does not require such large Fock states as input to the iterative scheme, which is advantageous because large Fock states are difficult to generate. By introducing some nondeterminism, large-amplitude Fock states can be guaranteed at the output for smaller Fock resource states. In fact, the disclosed method does not require Fock state resources. The disclosed method can be executed using arbitrary quantum states with even / odd photon number parity as input.

[0074] For example, the disclosed method can be supported by any resource state having dual phase rotation symmetry. In other words, the resource state only needs to have odd or even Fock parity (i.e., odd Fock parity means that the state consists only of an odd number of photon components, and the same applies to even Fock parity). The resource state can also be a squeezed Fock state. By squeezing the initial Fock state, more photons are deterministically placed in the final prepared Fock state. Another advantage of the disclosed method is that it uses a compressed vacuum instead of in-line compression, thus simplifying the entire process.

[0075] Although using smaller The value (i.e., a smaller Fock state) may introduce some nondeterminism—leading to not all states achieving high fidelity—which can be advantageous because it can selectively generate large-amplitude cat states from smaller Fock states, thereby enhancing the generality of the published method in producing high-quality cat states in experimental settings.

[0076] The cat state can also be used to deterministically prepare the GKP state, a process known as breeding. Therefore, this disclosure also provides a method for deterministically preparing the GKP state using the cat state prepared by the method and system described herein, and a system for this method.

[0077] In practice, losses and inefficiencies exist. However, this disclosure demonstrates that the disclosed method is robust to such noise and other defects and can be corrected for post-breeding errors using GKP correction. Numerical simulations are provided to validate the disclosed method. These simulations involve Monte Carlo simulations based on Fock under a Schrödinger picture. That is, they simulate the results of successive (shot-by-shot) experiments in actual experiments. For these simulations, in some Fock numbers... There exists an artificial truncation in the Fock space. This artificial truncation is large enough that the result cannot be changed by making the truncation larger, and the higher-order Fock components are very small.

[0078] This disclosure relates primarily to applications in quantum optical devices, such that the cat states and other states described by the disclosed methods are optical states or optical quantum states. However, it should be noted that the methods and systems described herein are not limited to optical applications. The methods and systems described herein are applicable to applications using general bosonic states and are not necessarily limited to optical bosonic states (i.e., photonic states). However, for simplicity and ease of interpretation, the methods and systems will generally be described in the context of quantum optical devices. Prerequisites

[0079] This disclosure generally relates to the use of Fock states, which may be referred to as number states or Fock number states. A Fock state is a quantum state that is an element of Fock space having a well-defined number of particles. This disclosure typically uses Fock states as single-mode Fock states. ,in This represents the number of particles described by the Fock state. In optical devices, these particles are photons; therefore, in the context of optical devices, the Fock state represents a photon-number state. (Creation operator) and annihilation operator Used to add particles to and subtract particles from the Fock state (i.e., and Vacuum state It is a quantum state without particles, in which creation and annihilation operators function in the following way: and .

[0080] Quantum states satisfy Heisenberg's uncertainty principle (or simply the uncertainty principle), which imposes a fundamental constraint on the product of the accuracy of certain related pairs of measurements on a quantum system (called observables or orthogonal components). The pair of orthogonal components exhibiting this uncertainty relation are position q and momentum p, to which the uncertainty principle gives... ,in It is the uncertainty (variance) of location, and This refers to the uncertainty (variance) of momentum. The variance of one orthogonal component can be reduced, at the expense of increasing the variance of another orthogonal component, because the uncertainty principle must still be satisfied even though the variances of each orthogonal component are not equal. The process of reducing the variance of an orthogonal component is called "squeezing," where the process of squeezing a quantum state results in a "squeezed state." For example, a quantum state can be squeezed using an optical resonator with a nonlinear crystal (such as lithium niobate or (periodically polarized) potassium titanate phosphate) placed between two or more mirrors.

[0081] One way to visualize quantum states (such as optical states) is to look at their phase space quasi-probability distribution, such as their Wigner function. The axis is determined by... The real and imaginary components of the complex amplitude are represented. Figure 1 The Wigner function (quasi-probability distribution of phase space) plots for the important single-mode optical states that will be considered in this disclosure are shown. Figure 1 In it, there exists a vacuum state 101 102 Compressed vacuum state with 10dB compression Fock state 103 (in It is an integer (in) Figure 1 middle, ), Two-component cat state 104 (in It is a coherent state), a compressed cat state 105 with 10 dB compression, and has a lattice spacing and 10dB compression ( The physical GKP state 106 (See Equation 5).

[0082] coherent state It is an annihilation operator : The eigenstates, which can be obtained by shifting the operator This displacement operator is applied to the vacuum state to create a state from the origin in the phase space (e.g., ...). Figure 1 (As shown) Displacement The vacuum state is a coherent state with zero amplitude. A coherent state is a state with minimal uncertainty, satisfying the lower bound of the uncertainty principle. The laser output has complex amplitude. coherent state ,in It is phase, and It refers to the absolute range.

[0083] The displacement and compression operators are defined as follows: and ,in It is the compression amplitude (i.e., the amount of compression), and It refers to the direction or angle of compression. (Real number) This represents displacement in a direction orthogonal to the position. and This indicates compression in a direction orthogonal to the position. For compression in a direction orthogonal to the position ( ) and reverse compression ( This can be written as .

[0084] Detection techniques such as homodyne detection and analytical photon number detection (PNRD) can be used during the preparation of cat states or GKP states. Homodyne detection is an optical interferometry that extracts information about the phase and / or frequency modulation of an oscillating signal encoded as an oscillating signal by comparing the oscillating signal to a standard oscillator that will be identical to the signal carrying null information. PNRD simply counts the number of individual photons in the signal. Homodyne detection projects onto orthogonal eigenstates, while analytical photon number detection (PNRD) projects onto the Fock state.

[0085] The phase rotation operator is ,in It is a rotation angle in radians. Phase rotation can be passively performed by causing the state to evolve over time.

[0086] In this disclosure, the objective is to prepare a compressed cat state in the following form: (1) (2)

[0087] here, It is a compressed vacuum state of displacement ( Take it as a real number), and It is a normalization constant. Essentially, the two-component cat state is a coherent state with opposite shifts. The sum of even or odd numbers (i.e., Even-numbered cat states have Superposition, and odd-numbered cat states have The superposition consists of only even or odd number of photon components. For example, for an even number of cat states: And for odd-numbered cat states: Compressed cat states are those with compression parameters. and displacement amplitude The cat state is a coherent superposition of the displacement of the compressed vacuum state. The prepared cat state can be corrected, for example, by single-mode compression and passive phase rotation, to obtain the correct amplitude and rotation required in Equation 1. For example, compression and passive phase rotation can be used to prepare the cat state as codewords for quantum error correction.

[0088] Qualitatively similar to the state in Equation 1 (i.e., by the state in...) The superposition of "spots" at a given location can be referred to as an approximate cat state or a cat-like state. Furthermore, if these "spots" appear compressed in the positional direction, these states can be called compressed cat-like states. In this disclosure, all cat-like states are simply referred to as cat states, and this imperfection can be arbitrarily small under large amplitudes because a significant amount of compression needs to be applied to the prepared cat states before breeding approximate GKP states, thus "compressing away" these imperfections. Moreover, due to the error-correcting properties of GKP codes, these imperfections can also be corrected during GKP breeding through GKP error correction.

[0089] In this disclosure, the following conventions are used to make it possible to define natural units Middle: Position And momentum The variances of the vacuum state and the coherent state are: The variance of the compressed vacuum state and the displacement of the compressed vacuum state is This convention is used for convenience; for example, in The expected value of the compressed vacuum state with displacement in the direction is only the amplitude. (i.e., no additional factors). In particular, inappropriate (unnormalized) orthogonal eigenstates. From the use of large The displacement of the compressed vacuum state To approximate.

[0090] Using this convention, the ideal square GKP logic ground state can be written as The position at the integer value - superposition of orthogonal eigenstates: (3) (4) in These are position eigenstates. These states can be considered as logical zero- and one-states that constitute a qubit, although in The phase space has built-in redundancy for error correction. Ideal GKP states are non-normalizable, have infinite energy, and are therefore non-physical.

[0091] One possibility for constructing physical GKP states is to replace positional eigenstates with squeezed vacuum states. That is, physical GKP states can be written as a superposition of vacuum states, by... Compress and displace from the origin Integer values. Furthermore, to handle infinite states, the envelope is applied to the amplitude, causing squeezed states far from the origin to be exponentially suppressed. This allows physically square GKP states to be defined as: (5) (6) Among them, envelope and compression parameters GKP compression in dB is given as Here, Equations 5 and 6 are finite-energy variations of Equations 3 and 4, respectively, where they asymptotically tend towards maximum compression. The same state.

[0092] A compressed cat state can be used as an approximate logic-GKP state, which consists of two compressed vacuum states with superimposed displacements of equal amplitude: (7)

[0093] Besides having only two most important central components In addition, this is Equation 6. The compressed cat state can be transformed into a physical GKP state with more components.

[0094] Quantum error correction using cat states and GKP

[0095] Quantum error-correcting codes will... Encoding 1 qubit into Redundancy is introduced into a subspace of the state space of a physical qubit (hereinafter referred to as the code space) in a way that allows for detection and error correction without interfering with the stored quantum information. More specifically, detection and error correction can be performed without measuring the quantum state itself, which will cause the quantum state to collapse based on the measurement result. Boson error-correcting codes encode a single logical qubit into a boson mode.

[0096] The code space is a two-dimensional subspace of the full Hilbert space, composed of encoding logic. and The two codeword states span a given state. As mentioned above, these two codewords can be composed of cat states, compressed cat states, and GKP states. Ideally, the codespace is defined such that when different errors (i.e., errors acting on the codewords) occur, each error results in a rotation to an error subspace, which is orthogonal to both the codespace and the error subspaces associated with other errors. Therefore, when an error occurs, quantum information is not lost but encoded in the error subspace. This allows for error detection and thus correction without degrading the quantum information stored in the code. Essentially, the error subspace can be measured, making it unnecessary to measure the codespace.

[0097] In optical applications, codewords can be stored in resonators and subjected to two main dissipation mechanisms: photon loss and pure dephase, which ultimately generate errors in boson quantum codes. Cat codes use either squeezed cat states (called squeezed cat codes) or unsqueezed cat states (called 2-cat codes).

[0098] Under large displacement constraints, quantum information encoded using 2-cat codes is autonomously protected against dephase errors. The suppression of dephase errors is due to the small overlap between the two coherent components of the cat state. Therefore, dephase errors are suppressed exponentially with the amplitude of the displacement field. Compacted cat codes offer additional protection against photon loss because they can function as GKP codes.

[0099] Therefore, this disclosure relates to a method for quantum error correction that uses a deterministically prepared cat state prepared using the method described herein. This quantum error correction method includes generating multiple codewords in a continuous-variable quantum space by applying one or more of a rotation, compression, shift, or parity measurement to the cat state prepared using the method described herein for each of a plurality of codewords. Applying one or more of a rotation, compression, shift, or parity measurement to the cat state enables the generation of appropriate codewords in 2-cat codes or compressed cat codes (or other cat codes).

[0100] Quantum information (such as the state of a qubit) is then encoded into multiple codewords to generate encoded quantum information. Finally, errors in the encoded quantum information are corrected based on the encoded quantum information or multiple codewords. Errors in the encoded quantum information can be corrected based on the encoded quantum information because both 2-cat codes and compressed cat codes autonomously correct dephase errors, while errors can be corrected based on multiple codewords by using multiple codewords as an aid.

[0101] GKP codes effectively prevent random displacement errors and losses in the phase space. To correct displacement errors in the phase space, GKP states are prepared as auxiliary states. Then, an entanglement gate (such as a SUM gate) is executed, with the auxiliary state as the control and the data as the target. Finally, the orthogonal components of the auxiliary state are measured. The results reveal the differences between the orthogonal measurements prior to the entanglement gate. The results are then modulo 2π / nα to determine the shift that should be applied to the data to recover from the error. To correct the displacement in the two orthogonal components, a two-state auxiliary circuit can be used, where each data point is entangled with each auxiliary state, and each auxiliary state is measured for one of the orthogonal components.

[0102] Therefore, this disclosure relates to a method for quantum error correction that uses a deterministically prepared cat state prepared using the method described herein. This quantum error correction method includes generating multiple codewords in a continuous-variable quantum space by applying one or more of rotation, compression, displacement, or parity measurements to the GKP state prepared using the method described herein for each of a plurality of codewords. Quantum information, such as a state of a qubit, is then encoded into the multiple codewords to generate encoded quantum information. Finally, errors in the encoded quantum information are corrected based on the encoded quantum information or the multiple codewords. Deterministic preparation of optical cat states

[0103] The result is given by performing the pass. The phase measurement at the origin, that is, similar to the phase measurement at an angle. The diameter of the circle is drawn, and the cat state is deterministically generated from the Fock state input via conditional measurements on the beam splitter. This is because the cat state has a known phase in phase space, and therefore, performing a phase measurement will cause the Fock state (which has an unknown phase in phase space) to collapse into the desired cat state. For a 50:50 beam splitter, the prepared cat state will have approximately 3 dB of compression and a known phase angle given by the measurement results. Rotate.

[0104] However, such a phase measurement cannot be performed. In optical devices, null detection and post-selection of the null result are equivalent to phase measurement; however, preparing the cat state via null detection is nondeterministic. Typically, null measurement (which can be considered as a projection onto an infinitely compressed state) does not pass through the center of the Fock state loop (as in...). Figure 1 (As can be seen in the Wigner function of the Fock state shown), this does not produce a cat state.

[0105] Here, a method and system for deterministically preparing two-component cat states are presented. The disclosed method uses an iterative approach, eliminating the need for back-selection and experimentally challenging feedforward mechanisms. This enables the preparation of cat states using simple equipment. The disclosed method is suitable for a large number of iterations. and all input photon number (Fock) states It is deterministic. Furthermore, the disclosed method eliminates the need for inline compression before the detector, making the method easier to implement.

[0106] Figure 2a A circuit for performing measurements using a compressed vacuum state as input is shown. Figure 2b A circuit for performing deterministic phase measurements is shown. The disclosed methods and systems behave like phase measurements, although direct phase measurements cannot be performed. Under the constraint of high transmittance, the phase probability distribution of arbitrary states is... Figure 2a and Figure 2b The two circuits shown evolve in the same way.

[0107] As mentioned above, Figure 2b The circuitry shown for deterministic phase measurement is not available in optics and other fields. However, compressed vacuum states (approximately orthogonal eigenstates) can be easily prepared offline, such as... Figure 2a The circuit shown is required. Therefore, by alternatively preparing a compressed vacuum state, the characteristics for performing phase measurements can be obtained.

[0108] As an example, consider ,in ,like Figure 2c As shown, and ,like Figure 2d As shown. Figure 2c It shows that for The evolution of the phase distribution of random input states using either a forward circuit (for preparing a rotating compressed vacuum) or a reverse circuit (for measuring a rotating compressed vacuum), and Figure 2d for The same content is shown. It can be seen that as long as... Both forward and reverse circuits exhibit similar phase evolution, allowing for the selection of a more practical circuit. Therefore, phase measurement can be simulated in an optical device (projected onto a rotating compressed vacuum) by alternatively preparing a compressed vacuum and projecting it onto the Fock state. Note that... This means that the transmittance should be as close to 100% as possible, but in reality it is not achieved, so that the minimum but non-zero amount of light can be reflected from the input state for photon detection.

[0109] More specifically, it can be seen that when At that time, the phase of the state has evolved in a similar way, but this applies to other transmittances. This is generally not the case. A forward circuit implies that a spin-compressed vacuum state is prepared and projected onto a Fock state, while a reverse circuit implies that a Fock state is prepared and projected onto a spin-compressed vacuum state. This suggests that, regardless of whether a compressed vacuum is input and a photon is measured, or a photon is input and a phase measurement is performed, the phase distribution of the state evolves similarly for high transmittance, with the latter being more difficult because a phase measurement cannot be performed. Then, through the former, it is as if a phase measurement exists in the optical device.

[0110] Figure 3a A system 300 for preparing the cat state 301 is shown. More specifically, Figure 3a A system for evolving a quantum state toward a cat state is illustrated. System 300 includes a beam splitter 310, which includes a first input 311, a second input 312, a first output 313, and a second output 314. System 300 also includes an initial input state source 320 for generating an initial input state 321, a squeezed vacuum source 330 for generating a squeezed vacuum state 331, and a detector 340 for detecting a measurement state 341. In some examples, detector 340 is a photon number analytic detector (PNRD) that counts the number of photons received at the first output 313. In other examples, detector 340 is a switch (threshold / bucket) detector, provided that the two-photon component of the initial input state is small enough to be negligible.

[0111] The initial input state source 320 may include various nonlinear crystals, source pumps (such as lasers), and a series of individual emitters (such as atoms, ions, molecules, nitrogen-vacancy (NV) centers, or quantum dots). In some examples, the initial input state source 320 is a source of Fock states, comprising a strong laser beam, referred to as a “pump” beam, directed toward a nonlinear crystal (e.g., a barium β-borate or lithium niobate crystal). This generates a pair of photons through a spontaneous parameter down-conversion process, wherein one photon in the pair is measured such that the measurement causes the other photon to collapse into a Fock state. This is merely one example of the generation of the initial input state 321. However, other generation methods are equally applicable. It should also be noted that the initial input state source 320 may include multiple Fock state sources and / or may be configured to generate multiple Fock states.

[0112] The compressed vacuum source 330 may also include various nonlinear crystals, source pumps (such as lasers), and a series of individual emitters (such as atoms, ions, molecules, nitrogen-vacancy (NV) centers, or quantum dots). In one example, the compressed vacuum source 330 may include an optical resonator with a nonlinear crystal (such as lithium niobate or (periodically polarized) potassium titanate phosphate) placed between two or more mirrors. Within the optical resonator, a process of optical parameter down-conversion (also known as optical parameter amplification) occurs. This allows the vacuum state to be transferred to a compressed vacuum state. This is just one example of generating the compressed vacuum state 331. However, other generation methods are equally applicable.

[0113] Figure 4 A method 400 for preparing a cat-like state is shown. (Reference) Figure 3b When the initial input state 321 is fed into the first input 311 and the initial compressed vacuum state 361 is fed into the second input 312, the beam splitter 310 creates an entangled state 401. The beam splitter 310 essentially mixes the initial input state 321 and the initial compressed vacuum state 361, thereby creating an entangled state with two output components. The compressed vacuum state 361 is determined by the compression direction (by...). Parameterization, such as Figure 1 As shown, the compression direction is the direction in the phase space where the compressed vacuum states 331 and 361 reside. The compression direction can be any direction (or any angle) in the phase space, and is not limited to any particular direction. Figure 1 direction shown.

[0114] The initial input state 321 is a quantum state with known Fock number parity. In some examples, the initial input state 321 is a quantum state with dual phase rotation symmetry. This means that the quantum state is symmetric in phase space when rotated by 180 degrees. An example of a quantum state with dual phase rotation symmetry is a single-mode Fock state, such as one that can be obtained from... Figure 1As seen in the diagram, it is infinitely symmetric in phase space because the Fock state corresponds to a circle. However, a quantum state with dual phase rotation symmetry can also be a quantum state with all odd or all even photon numbers.

[0115] In some examples, the initial input state source 320 can generate a first state as a quantum state, and the Fock number parity of the first state is measured. As the measurement collapses the quantum state into a state based on the measurement result, the measurement of the Fock number parity of the quantum state makes it possible to prepare an initial input state 321 with dual phase rotation symmetry. More specifically, a non-destructive measurement of the Fock number parity of any random high-photon state can create a useful initial input state with dual phase rotation symmetry for the disclosed method.

[0116] In other examples, the initial input state can be a combination of multiple quantum states, each of which has dual rotational symmetry. More specifically, a large-amplitude even / odd parity state can be a squeezed Fock state, or it can be constructed by combining even / odd parity units (single photons and squeezes) in an optical network.

[0117] refer to Figure 3b The detector 340 measures the entangled state 402 at the first output 313, which causes the entangled state to collapse into the output state 371 at the second output 314 of the splitter 310. In other words, performing a measurement on one mode projects another mode onto the quantum state based on the result at the detector.

[0118] Then, the process of creating entangled state 401 and measuring entangled state 402 is repeated. However, instead of using the initial input state to create the entangled state, the output state created by collapsing the previous entangled state is used as the input. Furthermore, the compression direction of the compressed vacuum state fed into beam splitter 310 is updated to a different compression direction. More specifically and referring to... Figure 3c The compressed vacuum source prepares an updated compressed vacuum state 381 by updating the compression direction. When the output state 371 is fed into the first input 311 of the beam splitter 310 and the updated compressed vacuum state 381 is fed into the second input 312 of the beam splitter 310, the beam splitter 310 creates a subsequent entangled state 404. The detector 340 then measures the subsequent entangled state 405 to generate a subsequent output state 391 at the second output 314 for feeding into the first input 311.

[0119] The process of preparing the updated compressed vacuum state (403), feeding the (subsequent) output state to the first input (311), and feeding the updated compressed vacuum state to the second input (312) to create the subsequent entangled state (404), and measuring the subsequent entangled state (405) to create the subsequent output state, is then repeated to prepare the cat state (406) at the second output (314). In some examples, the steps of creating and measuring the subsequent entangled state are repeated at least four times. In other examples, these steps are repeated 4 to 20 times in an inclusive sense. Essentially, these steps can be repeated as needed until the desired cat state is prepared; note that generally, the more repetitions, the better the resulting cat state.

[0120] In some examples, the initial input state 321 can be compressed before being fed into the first input 311. For example, the initial input state 321 can be a single-mode Fock state; therefore, compressing the initial input state 321 before being fed into the first input 311 makes the single-mode Fock state a compressed single-mode Fock state. This eliminates the requirement for such a large Fock photon number resource state by compressing the input Fock state. Generating a large Fock photon number state is difficult in practice; therefore, compressing the initial input state 321 before being fed into the first input 311 provides a simpler input for the disclosed method.

[0121] In some examples, the amplitude or compression amplitude can be increased by performing the following steps: creating an entangled cat state by feeding another cat state or a further compressed vacuum state prepared using the method described herein, along with the cat state, into an entanglement device; and measuring the entangled cat state at one output of the entanglement device using another detector, causing the entangled cat state to collapse into an improved cat state at another output of the entanglement device. Thus, the improved cat state has a larger amplitude or compression amplitude than the (initial prepared) cat state.

[0122] More specifically, the amount of compression (i.e., compression amplitude) and amplitude of the cat states prepared by the disclosed method can be increased by entanglement of two compressed cat states or a compressed cat state and a compressed vacuum state using a CNOT gate or beam splitter, and by measuring an output mode using null-difference detection to increase the compression and amplitude. This is similar to GKP breeding, as will be discussed later, but the relative compression orientation of each state and measurement can be different. For example, the measurement in the orthogonal component can be rotated relative to the usual case. .

[0123] Phase probability distribution can be used To track the phase of the states iteratively generated by the disclosed method. This requires an ideal gauge phase measurement, obtained by a positive operator. Given, where for , For the density matrix Obtain measurement results The probability is identified as Because the sum of probabilities is 1: .

[0124] Figure 5a , Figure 5c and Figure 6a , Figure 6c Numerical simulation results for deterministically preparing a two-component cat state from a Fock state are shown. This scheme is analogous to phase measurement, selecting two peaks from an arbitrary distribution of phase probability distributions. The simulation parameters are as follows: , The compressed vacuum has a compression of 6 dB. The transmittance of beam splitter 310... It should be very close to 1. In some examples, the transmittance of beam splitter 310 is between 0.9 and 1, making the total transmittance... Between 0.4 and 0.6 (and in some specific examples, around 0.5). The total transmittance is simply the transmittance of beam splitter 310. Number of iterations Power of 1.

[0125] Figure 5a The Fock state input is shown. Phase probability distribution The evolution of the phase probability distribution is shown, where the Fock state is not compressed. The evolution of the phase probability distribution from the earliest time to the latest time is illustrated. Figure 5b The initial Fock state is shown. The Wigner function. Figure 5c This demonstrates the use of Fock state input. The Wigner function for the cat state is prepared using the disclosed method, where the Fock state is not compressed.

[0126] Figure 6a The initial compressed Fock state with 6dB compression is shown. Phase probability distribution The evolution, in which . Figure 6b The initial compressed Fock state with 6dB compression is shown. The Wigner function, where . Figure 6c The initial compressed Fock state with 6dB compression is shown. The Wigner function for the cat state is prepared using the disclosed method, where... .

[0127] Figure 7a , Figure 7b and Figure 7cNumerical results for deterministically preparing two-component cat states from arbitrary input states are presented. The input states for this scheme are random states with odd Fock number parity and no more than 20 photons: Figure 7a The phase probability distribution of arbitrary input states is shown. . Figure 7b The Wigner function with arbitrary input states is shown. Figure 7c The Wigner function for the cat state prepared using the disclosed method with arbitrary input states is shown.

[0128] Compression occurs at different angles in each round. Vacuum input. It can be taken from the set of all discrete angles around the circle. For example, applied to... Compression of each mode in the pattern can proceed along different orientations. Guided. For values (i.e., iterations greater than 2), compression angle It can be set to be equidistant around a circle, specifically, ,in .exist In this case, it can be used .

[0129] However, random selection It is effective and simpler to implement. Especially for large... You can choose randomly. Thus, updating the compression direction in method 400 can include randomly selecting the direction. By randomly updating the compression direction during the iterative process of method 400, feedforward or inline compression is not required in the disclosed method. In some examples, the updated compression direction differs from each previously repeated compression direction.

[0130] The Fock number parity of the cat state prepared by method 400 is determined by parity. (If the initial state has even Fock parity) and parity (If the initial state has odd Fock parity) is given, where In the round The number of photons detected at that location.

[0131] The disclosed method deterministically prepares two-component cat states because the number of rounds is very large and From this point onward, the phase distribution smoothly approaches two peaks, which can be seen from... Figure 5a , Figure 6a and Figure 7a As can be seen from the phase probability distribution shown. Therefore, method 400 effectively performs phase measurement. The angle obtained therein Depends on all and The phase probability distribution smoothly approximates two peaks because a good approximation for the evolution probability distribution is: (8) Where the phase probability distribution Depends on the result at PNRD and input compression direction ,in It refers to the number of rounds. When the total number of rounds is large... hour, It is approximated as a cosine function. In this case, a good approximation is given when the result is even. When the result is odd, it is ,in This is the result of an effective phase measurement. That is to say, in... At the wheel, if the result at PNRD is odd and the parity of the evolved state changes, then the phase probability distribution changes significantly.

[0132] For multiple iterations of simulation It always ends up in and Two strong peaks are obtained at this point, indicating that it definitively evolves towards a two-component cat state. Therefore, the effective phase measurement results are... The values ​​are taken as completely random and begin with a constant-phase probability distribution. . Actual Implementation

[0133] An example embodiment is now provided, which is a simplified implementation of the disclosed method for deterministically preparing cat states, a method that can be carried out using existing techniques. First, the requirement for large Fock states is reduced. Any input state with even or odd Fock number parity can be the initial input state for the disclosed method. Therefore, compression can be used to reduce the size requirement of Fock number resources. The trade-off between the success probability and the size of the obtainable cat states is... Figure 8a It is qualitatively shown in the text.

[0134] Figure 8a The relationship between the success probability and the Fock number cutoff is shown, indicating the probability of success for a compressed Fock state with 10 dB compression. The source, the expected cat amplitude, and the achievable success probability. Each curve is for a different initial number of Focks. y The axis plots the compressed Fock state, which has more than [a certain value]. x The probability of the cutoff photon number is shown on the axis. The size of the cutoff approximates the squared magnitude of the obtainable cat states. And the compression of the GKP state in breeding. dB, as given in Equation 9. This figure illustrates that large cat states, and therefore highly compressed GKP states, can be achieved with a lower success probability, but with a higher probability for larger initial Fock numbers. Although since the initial compression is fixed and not included in... Figure 8a As shown, the probability is also higher for more initial compression on the initial Fock state.

[0135] Numerical simulations show that, by visually comparing Wigner portraits in phase space, for Wheel, 6dB compressed vacuum source and Total transmittance can be used to prepare materials with amplitude. The qualitatively well-defined two-component cat state has a probability of approximately 10%, which is from... Figure 8a As expected.

[0136] As compression parameters The average number of photons in a compressed Fock state as a function of the Fock state compression (dB) exist Figure 8b The figure illustrates the average number of photons available at the input of the disclosed method for preparing the cat state, resulting in a larger prepared cat state as a function of Fock state compression. The figure demonstrates the significant advantage of using compressed single-photon input instead of compressed vacuum input.

[0137] Average fidelity and compression

[0138] Since the prepared cat forms are intended for use in creating GKP forms, the fidelity and compression of the prepared cat forms can be calculated to determine how good the compressed cat forms are for GKP breeding.

[0139] Fidelity is a measure of the "closeness" between two quantum states. In this sense, fidelity is used as a comparison metric between the prepared output state and the ideal target cat state. Given the variability of the output state's amplitude, parity, and squashing, the prepared cat state can first be aligned onto a GKP grid using Gaussian operations. As described in Equation 7. This may be an ideal configuration for GKP breeding. To obtain the ideal compression level of the target cat form, examine the prepared cat form's... The variance of orthogonal components.

[0140] The average fidelity between the corrected output cat state of the disclosed method and the target compressed cat state from Equation 7 is calculated, where the compression amount is selected to match the output cat state, based on the output cat state. The variance of the orthogonal components is numerically determined. The expected compression is given by the following equation: (9).

[0141] Figure 9a and Figure 9b The quality of the compressed cat state prepared using the disclosed method is shown. Figure 9a This shows the effect of a fixed number of Fock states. As the total transmittance The average fidelity of the disclosed method for the function in 50 experiments. Total number of rounds. (That is, large) and the compressed vacuum resource state has a compression of 6dB. From Figure 9a As can be seen, fidelity decreases with increasing total transmittance.

[0142] See Figure 9a and Figure 9b The non-smooth behavior in the figures is due to the existence of many possible sequences of outcomes during the simulation. Averaging the results over more simulation runs may smooth these curves. Furthermore, the simulations are only approximations due to the numerical truncation of the Hilbert space. However, the overall trends in compression and fidelity can be easily observed in these figures.

[0143] Figure 9b It shows that for Figure 9a The same parameter in the total transmittance is used as the total transmittance. The compression of the function in dB (numerically derived from...) (variance in the data). Figure 9b An analytical approximate compression (straight line) was also plotted. From Figure 9b As can be seen, compression increases with total transmittance. This implies a trade-off between fidelity and compression, which can be adjusted by changing the total transmittance. Therefore, the disclosed method is suitable for larger... It can still work well, which means it produces more compression. In some examples, Being chosen as large, therefore for moderately small There will be a relatively large amount of compression, followed by some post-selection to remove "bad" cat states. In other words, average fidelity can be improved by only performing post-selection on the best compressed cat states.

[0144] The results reveal a trade-off between fidelity and compression. Notably, these results can be enhanced through selective post-processing of the output, focusing on the highest-quality cat states generated during the experimental run. It is estimated that, assuming a detector efficiency of 98%, for This ensures that the disclosed method achieves good fidelity in experiments. k Good practical values ​​are between 4 and 10. More details and how average fidelity varies with [the value is missing from the original text] are shown later in this disclosure. k The changing curve (see Figure 17 ).

[0145] Improve the scheme with feedforward operation

[0146] Although the disclosed method deterministically prepares cat states without the need for feedforward techniques, these feedforward techniques have the potential to significantly improve the disclosed method for practical applications, which could lead to enhanced cat states or simplify the entire method.

[0147] In simple terms, by identifying an “undesirable situation” after only a few rounds, the beam splitter ratio can be adjusted so that the process can restart and the state evolves as expected. In other words, if the state does not evolve as expected, the beam splitter transmittance is adjusted to allow more compressed vacuum to enter, and the protocol continues as normal. Therefore, in some examples, method 400 may also include adjusting the beam splitter transmittance based on the output state or subsequent output states during repetition.

[0148] In some examples, the compressed vacuum state can be determined by the compression amplitude or intensity (by...). The compression amplitude can be parameterized using a representation, and more complex feedforward methods can target a specific compression direction (angle) or compression intensity based on previous measurements. Therefore, method 400 can further include updating the compression amplitude based on the result of measuring the entangled state during repetition or the result of measuring a subsequent entangled state. Furthermore, updating the compression direction can include updating the compression direction based on the result of measuring the entangled state during repetition or the result of measuring a subsequent entangled state. GKP state preparation

[0149] Figure 10 A system 1000 for preparing GKP state 1001 is shown. The GKP state can be bred from a deterministic source of compressed 2-cat state. System 1000 includes... Figure 3a System 300 uses method 400 to prepare cat states. System 1000 also includes an entanglement device 1010 and a detector 1020. The entanglement device 1010 includes inputs 1011 and 1012, outputs 1013 and 1014. In some examples, the entanglement device 1010 is a CNOT logic gate or a beam splitter, and the detector 1020 is a zero-difference detector configured to perform zero-difference momentum detection. Note that the detector 1020 can be other detectors, such as a zero-difference detector configured to perform zero-difference position detection.

[0150] Figure 11 A method 1100 for preparing GKP state 1001 is shown. System 300 uses, as follows: Figure 4The method 400, as shown and described above, prepares 1101 a first cat state 1031 and a second cat state 1032. The entanglement device 1010 then creates 1102 a further entangled state as the first cat state 1031 and the second cat state 1032 are fed into the entanglement device 1010. More specifically, the first cat state 1031 is fed into input 1011 of the entanglement device 1010, and the second cat state 1032 is fed into input 1012. The detector 1020 then measures 1103 the further entangled state at output 1013 of the entanglement device 1010 to cause the further entangled state to collapse into a GKP state 1001 at output 1014.

[0151] The deterministic breeding of GKP states from compressed two-component cat states can be explained by considering a quantum gate circuit model in code space. Compressed cat states form primitive logic-GKP states. By applying the GKP CNOT gate Two copies of the state, after the gate, the second mode will undergo a bit flip and is now... This time it has three peaks. Through bibasal (zero difference) The first mode is measured in the input state, which ensures that the output state is pure. Therefore, the breeding of the GKP state is bred to have three peaks. This process can be iterated to combine the squeezed cat state and the bred GKP qubit, thus adding more squeezed states to the superposition at the output.

[0152] Preferred embodiments for preparing the GKP state will now be described. Figure 12 It shows Figure 10 The system shown is a preferred embodiment. In this preferred embodiment, the input cat states 1231 and 1232 prepared using system 300 are first corrected by correction operations 1241 and 1242 (such as passive phase rotation and compression) to make the amplitude and compression equal. In direction, and in magnitude In other words, method 1100 further includes correcting the first and second cat states before creating further entangled states by rotating and compressing the first and second cat states. In this embodiment, the entanglement device is a CNOT logic gate 1210 (SUM= This causes the two modes to become entangled, and one mode is measured by detector 1220, which is Zero-difference detector in orthogonal components.

[0153] More specifically, in a preferred embodiment, detector 1220 is a zero-difference detector configured to perform zero-difference detection of orthogonal components perpendicular to the compression direction or amplitude direction of the first cat state or the second cat state. Following detection, the GKP state is then prepared in the second mode.

[0154] Instead of using the cat state prepared using method 400 as input to the entanglement device 1010, further rounds can use system 1000 to breed a larger GKP state. Therefore, method 1100 may include repeating the steps of method 1100 to prepare a second GKP state, and breeding the GKP state and the second GKP state to prepare a higher-component GKP state. If two three-component GKP states are used as input, the higher-component GKP state can be a five-component GKP.

[0155] Figure 13 Numerical results for GKP breeding from cat forms prepared using the methods disclosed herein are shown. The format is as follows: (in The source is any large-amplitude input state 1301, 1302, 1303, 1304 (which is any random number) to execute. Figure 13 In breeding, where the larger the Fock number, the greater the weight is determined by Provided. Figure 13 Two rounds of breeding are shown, in which cat states were prepared and then randomly bred. Two compressed cat states, 1311 and 1312, were bred to give a three-peaked GKP state, 1321. Similarly, two compressed cat states, 1331 and 1332, were bred to give a three-peaked (three-component) GKP state, 1341. Two GKP states, 1321 and 1341, were bred to give a five-peaked (five-component) GKP state, 1351.

[0156] To successfully breed GKP states from cat states, the primordial cat state requires high amplitude and / or compression, such that the cat state should have sufficiently high compressibility once compressed to the desired spacing to prepare the GKP state. This can be achieved by using a large initial cat amplitude (before compression). This is done so that the cat state is highly compressed after compression. It can take any value because the cat state can be compressed to the correct spacing. To successfully breed GKP states, a positive peak at the origin of the Wigner distribution is required. For odd-numbered cat states, this can be achieved by... Small displacements in the orthogonal component directions are used for repair.

[0157] Simple calculations provide the minimum Fock number required to realize high-quality GKP states for universal fault-tolerant quantum computing. If quantum gates are used, the source of the compressed cat states used for breeding is... If a bundle splitter is used in breeding, this reduces the factor. ,in z It refers to the number of breeding rounds. It is also an estimated number of photons used to achieve specific compression and quality in the output GKP state. This can be found in Equation 9, which means that we need to use... , To obtain 10dB ( ).

[0158] Big Under certain constraints, ideal-quality big-cat states can be prepared, which can be bred to produce high-quality approximate GKP states. Therefore, fault-tolerant quantum computing can be performed using GKP error correction. To further reduce the Fock number, the transmittance of the beam splitter can be selected to make... Therefore, the output has a larger amplitude and greater compression of the cat state. However, fidelity will be affected, such as... Figure 9a As shown. GKP error correction performance

[0159] Before the GKP state can be deterministically prepared through breeding, the prepared cat state can be adjusted to the form given by Equation 7. For a given total transmittance Equation 9 is established and The relationship between them. By setting... Observed as Increase, compression increases size and fidelity approaches one, that is and Targeting large The high fidelity can be attributed to the reduction in phase uncertainty in the effective phase measurement performed. Therefore, approximate GKP states can be prepared with particularly high quality, characterized by fidelity and compression. This enables fault-tolerant quantum computing using GKP error correction via the disclosed state preparation method.

[0160] The fidelity and compression calculations described above illustrate the usefulness of the prepared cat states. This analysis is now extended to GKP states to verify the goodness of the prepared states for error correction.

[0161] Average optimal entanglement fidelity

[0162] Consider preparing compressed cat states and randomly combining two of them to breed GKP states (one breeding level, so that the GKP state has three peaks). Then, use two of these randomly prepared GKP states to define the basis of the logic code. The prepared GKP code is defined as follows: and Half of a qubit Bell pair is encoded into a code. Loss is applied to the encoded qubit (which is the primary noise source in the optical device), optimal error correction is applied for recovery, and then the code is decoded back into qubit space. The fidelity of the output entangled pair with the initial Bell pair is then calculated. This defines the entanglement fidelity resisting loss for a given preparation code. The optimal entanglement fidelity is obtained using CVX in MATLAB with a semi-definite procedure (SDP). This is calculated over several rounds of experiments to obtain the average optimal entanglement fidelity. That is, two random cat states are combined to prepare a logic-zero GKP state, and different random cat state pairs are combined to prepare a logic-one state.

[0163] Figure 14 Numerical simulation results of GKP state preparation using the disclosed method are shown, assuming that the source of the input Fock state can be used for cat state preparation and a round of GKP breeding from the compressed cat state. The average optimal entanglement fidelity is plotted as the initial photon number. The function, assuming the encoded entangled Bell pairs are subjected to a transmissivity The pure loss error channel is affected, and then the optimal recovery operation is applied. The results are averaged across 10 experiments (only rough results are given).

[0164] Figure 14 The results were averaged across 10 complete simulations of the GKP breeding experiment. The results also show the results derived from trivial codes. and (As shown by horizontal line 1401) and by and The given trivial compressed coherent state codes (as shown by horizontal line 1402) provide the break-even point for error correction, where... and Furthermore, the compression is 10 dB. It can be analytically demonstrated, and the figure numerically confirms, that if the amplitude of the input state is sufficiently large, the GKP state prepared using the disclosed method can achieve a break-even point for error correction. The figure illustrates how the cat-state and GKP state preparation schemes are robust to loss because GKP error correction is used to correct for the loss.

[0165] Note that these results are approximations because the Hilbert space is numerically truncated. In any case, according to Figure 14 When n is greater than or equal to about 4, the GKP state preparation scheme appears to become useful for error correction. loss

[0166] In reality, due to imperfect systems and noise, there are losses and low efficiency. Here, the pure loss channel is placed directly in a channel with transmittance... Before the detector, there is an ideal detector, in order to achieve efficiency. Modeling imperfect detectors.

[0167] To see how the maximum allowable loss is proportional to the size of the input Fock state, the average fidelity of 50 trials is [value missing]. Figure 15 The input Fock state is drawn in the middle. The function, where Each curve has a different detector efficiency value. Total transmittance is In this example, the input Fock state is uncompressed, and the compressed vacuum has a compression of 6 dB. The average fidelity is observed to increase with... The decrease is proportional to the decrease in the number of input photons, and notably, it also decreases with the decrease in the number of optimization iterations. and total transmittance It can enhance the targeting of specific The average fidelity.

[0168] Figure 15 The different loss amounts before each detector are shown (i.e., detector efficiencies as illustrated in the legend). ) fidelity and input Fock state The relationship between the total transmittance and the total transmittance. Furthermore, the Fock input state is not compressed. Note that these results are presented to illustrate the effect for many different... With low noise levels, good fidelity is possible, but there's no need to be overly concerned, because... and These are experimentally available parameters that can be selected to optimize fidelity; however, for simplicity, they are fixed here. Note that choosing optimal experimental parameters, such as the number of iterations, is crucial. Total transmittance The degree of compression can significantly enhance the performance for a specific initial Fock state. and detector efficiency The average fidelity. Non-ideal 2-fold rotational symmetry

[0169] Due to experimental limitations, high-purity input states may be impractical. However, even if high-purity input states do not exhibit exact 2-fold rotational symmetry (and therefore exact even or odd photon number parity), non-Gaussian cat-like features can still be observed. To illustrate this, consider the input state as having the form Examples of cat states, where In this example, we assume the following factor: total number of iterations. Ideal detector, compressed vacuum with 6dB compression, and total transmittance The result of this example is... Figure 16a and Figure 16b As shown in the image.

[0170] Figure 16a A graph of the Wigner function is shown, illustrating an ideal input state consisting of cat states (i.e., a pure state with exact double rotation symmetry) and a typical output state for the ideal input state. Figure 16b Plots of the Wigner function for a non-ideal input state (i.e., a pure state that is not exactly double rotationally symmetric) and for the corresponding typical output for a non-ideal input are shown. Interference fringes are still observed in the output, so the slight defect in rotational symmetry at the input still results in a cat-like state with non-Gaussian characteristics at the output (in this example, the output looks like an unbalanced cat state). It can be seen that the generated output cat state is worse when the input state is not exactly double rotationally symmetric; however, the non-Gaussian characteristics still exist. Optimal number of iterations

[0171] To estimate the number of iterations required to achieve a certain fidelity, a method for the disclosed approach using both ideal and non-ideal detectors was generated. A graph of the fidelity of the function. Figure 17 Showing the target The input Fock state, assuming an ideal detector (circular marker) and efficiency of In the case of an inefficient detector (fork marker), the average fidelity The number of iterations compared to the prepared compressed cat state The relationship is shown in the figure, illustrating the fidelity. The results are averaged across 50 simulations. This configuration involves a compressed vacuum subjected to 6dB compression, with a total transmittance of [missing information]. .

[0172] The disclosed method using ideal detectors, along with The performance improves with the increase of [something]. However, for non-ideal detectors, the published method [performs better]. It shouldn't be too large, because more detectors add more noise to the final state. Number of iterations It should be between approximately 4 and 10, and should be at least... It produces good experimental results.

[0173] Those skilled in the art will understand that many variations and / or modifications can be made to the above embodiments without departing from the broad general scope of this disclosure. Therefore, these embodiments are to be considered illustrative rather than restrictive in all respects.

Claims

1. A method for preparing a cat-like state, the method comprising: An entangled state is created by feeding an initial input state into the first input of the beam splitter and an initial compressed vacuum state into the second input of the beam splitter, the compressed vacuum state being parameterized by the compression direction; The entangled state is measured at the first output of the beam splitter using a detector, so that the entangled state collapses into the output state at the second output of the beam splitter; The subsequent entangled state is created in the following way: The output state is fed into the first input; Update the compression direction to prepare a new compressed vacuum state; as well as The updated compressed vacuum state is fed into the second input; The subsequent entangled state is measured to generate a subsequent output state for feeding into the first input; as well as The cat state is iteratively prepared at the second output by repeating the steps of creating and measuring the subsequent entangled states.

2. The method according to claim 1, wherein, The initial input state is a quantum state with dual phase rotation symmetry.

3. The method according to claim 2, wherein, The method further includes measuring the Fock number parity of the first state to prepare the initial input state having dual phase rotational symmetry.

4. The method according to claim 2, wherein, The initial input state is a combination of multiple quantum states, each of which has dual rotational symmetry.

5. The method according to claim 1 or 2, wherein, The initial input state is a single-mode Fock state.

6. The method according to claim 5, wherein, The single-mode Fock state is a compressed single-mode Fock state.

7. The method according to any one of the preceding claims, wherein, The method further includes adjusting the transmittance of the beam splitter based on the output state or the subsequent output state during repetition.

8. The method according to any one of the preceding claims, wherein, The compressed vacuum state is parameterized by the compression amplitude; and The method further includes updating the compression amplitude based on the result of measuring the entangled state during repetition or the result of measuring the subsequent entangled state.

9. The method according to any one of the preceding claims, wherein, Updating the compression direction includes randomly selecting a direction.

10. The method according to any one of the preceding claims, wherein, Updating the compression direction includes updating the compression direction to a direction different from each previously repeated compression direction.

11. The method according to any one of claims 1 to 8, wherein, The compression direction is updated based on the result of measuring the entangled state during the repetition or the result of measuring the subsequent entangled state.

12. The method according to any one of claims 7 to 11, wherein, The transmittance of the beam splitter is between 0.9 and 1, resulting in a total transmittance between 0.4 and 0.

6.

13. The method according to any one of the preceding claims, wherein, The steps of creating the subsequent entangled state and measuring the subsequent entangled state are repeated at least four times.

14. The method according to any one of the preceding claims, wherein, The cat state and the initial input state are optical states.

15. The method according to any one of the preceding claims, wherein, The method further includes increasing the amplitude of the cat state or the compression amplitude by performing the following steps: An entangled cat state is created by feeding one of a cat state or another compressed vacuum state prepared using the method according to any one of the preceding claims, along with the cat state, into an entanglement device; and The entangled cat state is measured at one output of the entanglement device using another detector, causing the entangled cat state to collapse into an improved cat state at the other output of the entanglement device. The improved cat state has a larger amplitude or compression amplitude than the original cat state.

16. A method for quantum error correction, the method comprising: For each of the plurality of codewords, the plurality of codewords are generated in a continuous variable quantum space by applying one or more of rotation, compression, displacement or parity measurement to a cat state prepared using the method according to any one of the preceding claims; Quantum information is encoded into the plurality of codewords to generate encoded quantum information; as well as Errors in the encoded quantum information are corrected based on the encoded quantum information or the plurality of codewords.

17. A method for preparing a Gottesman Kitaev-Preskill (GKP) state, the method comprising: The first cat state and the second cat state are prepared using the method according to any one of claims 1 to 15; Another entangled state is created by feeding the first cat state and the second cat state into another entanglement device; and The other entangled state is measured at one output of the other entangled device using another detector, so that the other entangled state collapses into the GKP state at another output of the other entangled device.

18. The method according to claim 17, wherein, The method further includes correcting the first cat state and the second cat state before creating the other entangled state by rotating and compressing the first cat state and the second cat state.

19. The method according to claim 17 or 18, wherein, The method further includes correcting the GKP state by rotating and compressing it.

20. The method according to any one of claims 17 to 19, wherein, The method further includes: Repeat the method according to any one of claims 17 to 19 to prepare the second GKP state; and Breeding is performed on the GKP state and the second GKP state to prepare a GKP state with a higher content.

21. A method for quantum error correction, the method comprising: For each of the plurality of codewords, the plurality of codewords are generated in a continuous variable quantum space by applying one or more of rotation, compression, displacement or parity measurement to the GKP state prepared using the method according to any one of claims 17 to 20. Quantum information is encoded into the plurality of codewords to generate encoded quantum information; as well as Errors in the encoded quantum information are corrected based on the encoded quantum information or the plurality of codewords.

22. A system for preparing a cat-like state, the system comprising: The initial input state source is configured to generate the initial input state; A compressed vacuum source is configured to generate an initial compressed vacuum state and an updated compressed vacuum state, wherein the initial compressed vacuum state and the updated compressed vacuum state are parameterized by a compression direction; A beam splitter includes a first input, a second input, a first output, and a second output, wherein the beam splitter is configured to create entangled states when feeding the initial input state to the first input and feeding the initial compressed vacuum state to the second input; and The detector is configured to measure the entangled state at the first output to cause the entangled state to collapse into an output state at the second output; in The beam splitter is also configured to create subsequent entangled states when feeding the output state to the first input and feeding the updated compressed vacuum state to the second input; The detector is also configured to measure the subsequent entangled state to generate a subsequent output state for feeding into the first input; and The cat state is iteratively prepared at the second output by repeating the steps of creating and measuring the subsequent entangled states.

23. A system for preparing GKP states, the system comprising: The system according to claim 21 or 22 is configured to prepare a first cat state and a second cat state; Another entanglement device is configured to create another entangled state while feeding the first cat state and the second cat state; as well as Another detector is configured to measure the other entangled state at one output of the other entanglement device, such that the other entangled state collapses into the GKP state at another output of the entanglement device.

24. The system according to claim 23, wherein, The other entanglement device is a CNOT logic gate or a beam splitter.

25. The system according to claim 23 or 24, wherein, The other detector is a zero-difference detector, which is configured to perform zero-difference detection of orthogonal components perpendicular to the compression direction or amplitude direction of the first cat state or the second cat state.

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