Systems, methods, and control instructions for simultaneously preventing bit flip and phase flip error types of a quantum subsystem

Abstract

The system can include: a quantum subsystem drivable to carry a bosonic cat state having N coherent states, where N is greater than 2, the N coherent states defining a computational space; a controller operatively connected to control the driving hardware according to control instructions, where at least one type of interaction between the bosonic cat state and an environment not originating from the control instructions causes a phase shift of the N coherent states, where the control instructions include a function to generate an initial state of the bosonic cat state defined as a given superposition of two different logical states representing logical information, the two different logical states spanning a logical subspace within the computational space, where any phase shift from the initial state results in a modified state that lies outside of the logical subspace while preserving the logical information or returns to the initial state.

Description

Technical Field

[0001] This application relates to systems, methods, and control instructions for simultaneously preventing bit-flipping and phase-flipping error types in quantum subsystems. Background Technology

[0002] In recent years, quantum computing has become an active research field due to its potential to revolutionize modern computing, communication, and cryptography.

[0003] In contrast to classical computers, where information is physically stored in memory in binary form (bits), retrieved, transmitted, and processed, in quantum computers, information is physically stored in memory by encoding it as the state of quantum subsystems (qubits), and can be processed by quantum algorithms and measured (retrieved) based on von Neumann entropy.

[0004] One of the main challenges in encoding quantum information in practical physical devices is balancing two conflicting requirements. First, quantum information is inherently fragile and susceptible to destructive decoherence sources. Unfortunately, most interactions with the environment are decoherence sources. Therefore, systems designed to store and process quantum information are typically isolated from interactions with the environment as much as possible. Second, for quantum information to be useful, it needs to be processed and eventually accessed, which means controlling the quantum information through interactions with an external control system. To make the processor efficient, these control interactions must be performed at relatively high frequencies.

[0005] Because of this second requirement, errors in quantum information processing are difficult to avoid. A common approach inspired by classical information processing attempts to partially correct these errors through redundancy, such as redundantly encoding quantum information at additional physical sources. This approach is prohibitively expensive, with typical overhead on the order of 10,000 times for practical applications, making it unattractive to many real-world applications.

[0006] A newer approach is to encode qubits in an already richer quantum subsystem in a way that allows quantum information to be redundantly encoded within a single physical element. A promising avenue for this approach is boson encoding, where the phase space of a single system can provide a theoretically infinite-dimensional Hilbert space to encode information.

[0007] In an infinitely large encoding space, there are theoretically an infinite number of ways to encode qubits. In practice, a few promising candidates with interesting properties have recently emerged. One such candidate is the so-called cat-state encoding, where the qubit is encoded in the superposition of two quasi-classical coherent states of the system. Although recent research on cat-state encoding has yielded interesting results, further improvements are still needed. In particular, it remains necessary to simultaneously prevent bit-flip and phase-flip error types. Summary of the Invention

[0008] According to one aspect, a method is provided for carrying a boson cat state having N coherent states in a quantum subsystem, where N is greater than 2, the N coherent states defining a computational space. The method includes generating an initial state of the boson cat state, which is a given superposition of two distinct logical states representing logical information, the two distinct logical states spanning a logical subspace within the computational space. The two logical states are defined such that any phase shift of the boson cat state from the initial state, originating from an uncontrolled interaction between the boson cat state and its environment, is limited to causing one or more modified states located outside the logical subspace that simultaneously retain the logical information, or returning to the initial state.

[0009] According to another aspect, a quantum system is provided, comprising: a quantum subsystem drivable to carry a boson cat state having N coherent states, where N is greater than 2, the N coherent states defining a computational space; driving hardware operatively connected to drive the quantum subsystem to carry the boson cat state; and a controller operatively connected to control the driving hardware according to control instructions, wherein at least one type of interaction between the boson cat state and the environment, not derived from the control instructions, causes a phase shift in the N coherent states, wherein the control instructions include a function generating an initial state of the boson cat state, the initial state being defined as a given superposition of two distinct logical states representing logical information, the two distinct logical states spanning a logical subspace within the computational space, wherein any phase shift from the initial state results in a modified state located outside the logical subspace that simultaneously preserves the logical information, or a return to the initial state.

[0010] According to another aspect, there is provided a control instruction stored in a non-transitory computer-readable memory, the control instruction comprising: a function, when executed by a computer, generating an initial state of a boson cat state having N coherent states in a quantum subsystem, where N is greater than 2, the N coherent states defining a computational space, the initial state being defined as a given superposition of two distinct logical states representing logical information, the two distinct logical states spanning a logical subspace within the computational space, and wherein at least one type of interaction between the boson cat state and the environment, not originating from the control instruction, causes a phase shift in the N coherent states; and a definition of the two distinct logical states, the definition restricting the boson cat state to either evolve into a modified state located outside the logical subspace while retaining the logical information, or return to the initial state, at any phase shift from the initial state.

[0011] According to another aspect, a quantum system is provided, comprising: a quantum subsystem drivable to carry a cat state having N coherent states, the N coherent states being representable as points around a circle in the phase space of the quantum system, where N is greater than 2; driving hardware operably connected to drive the quantum subsystem to carry the cat state by controlling a plurality of bosons in the quantum system; and a controller operably connected to control the driving hardware in a manner that encodes two distinct logic states with the N coherent states, wherein the logic states are rotationally asymmetric as the cat state rotates around the circle due to boson loss in the quantum system.

[0012] According to another aspect, a method for performing quantum processing is provided, comprising: driving a cat state having N coherent states in a quantum subsystem, the N coherent states being representable as points around a circle in the phase space of the quantum subsystem, where N is greater than 2; the driving comprising controlling a plurality of bosons in the quantum system to encode two different logic states with the N coherent states, wherein the logic states are rotationally asymmetric when the cat state is rotated around the circle due to boson loss in the quantum system.

[0013] Upon reading this disclosure, many further features and combinations thereof related to this improvement will be apparent to those skilled in the art. Attached Figure Description

[0014] In the attached diagram:

[0015] Figure 1 shows a more detailed phase space representation of a) 2-cat state, b) 4-cat state, and c) 3-cat state, illustrating the superposition of coherent states (great circles) and the interference fringes they produce;

[0016] Figure 2 It is the meta-potential representation of Hamiltonian stabilization;

[0017] Figure 3 This is a schematic diagram of a quantum system that can be implemented as a quantum processor;

[0018] Figure 4 This is a schematic diagram illustrating the cleaning operation process in a step-by-step manner;

[0019] Figure 5 This is a schematic illustration of the modes and interactions that enable the logic qubits embedded in the three-legged cat state (3-cat) to achieve stabilization and quantum error correction;

[0020] Figure 6 This is a schematic diagram of an example circuit used to implement the proposed protocol; and

[0021] Figures 7a to 7j It is a set of schematic diagrams representing N individual states in phase space, where the N states can be associated with logical states and form the basis of quantum bit encoding.

[0022] Figures 8a to 8e and Figures 9a to 9f This is a schematic representation of the phase evolution due to the loss of consecutive bosons, defined for two example logic states. Detailed Implementation

[0023] The quantum states of bosons in a resonator can be modeled as a quantum harmonic oscillator system. Any possible state of this system can be described by a distribution of states at a specific excitation energy level (called the bosonic number state basis, or Fock state basis). These states can also be represented by a quasi-distribution in two-dimensional phase space given by the Wigner transform of the wavefunction of the state.

[0024] The lowest energy state is called the vacuum state. In phase space, its Wigner transform is represented by a circular quasi-Gaussian distribution at the origin of the two orthogonal quadratures of the Wigner transform.

[0025] The result of applying the displacement operator to the vacuum state is defined as a coherent state (whose representation in phase space is the same as that of the vacuum state) until translation.

[0026] This displacement can be parameterized by a complex number α, with its real part describing the horizontal orthogonal displacement and its imaginary part describing the vertical orthogonal displacement.

[0027] Coherent states with the same modulus and α value all lie on a circle with radius |α|, and they all represent states with the same average boson number (while... <n>=|α| 2 ).

[0028] A cat state can be defined as any superposition of any number of coherent states located on a circle centered at the origin of phase space, and thus sharing the same |α| value.

[0029] In particular, an n-cat state is formed by the superposition of n uniformly distributed states along the circumference of a circle with radius |α|.

[0030] Figure 1a An example of a quantum subsystem carrying a cat state is given, where the cat state has two coherent states represented as equidistant points around a circle in phase space. This cat state can be simply called a 2-cat.

[0031] Using a stabilization mechanism leveraging an external EM driver, the 2-cat state achieves a long lifetime. This encoding is robust to so-called bit-flip errors (where a pure state |0> will become |1>, and vice versa). However, under the influence of environmental excitation losses (e.g., single-photon losses), this encoding may not be robust to another important error channel, phase flips (where a given state superposition (e.g., |0>+|1>) will transform into a different superposition (e.g., |0>-|1>)). This leaves a need for physical redundancy, even though this need can be mitigated by addressing bit-flip type errors, one of the two main error channels. Therefore, it remains necessary to address both bit-flip and phase-flip error types simultaneously.

[0032] When the system state following such unwanted interaction also represents a valid logical state, a possible source of error in the boson cat state is single-photon loss. In this case, it may not be possible to detect the error simply from the logical state. That is, for a general qubit state given by:

[0033]

[0034] Where u is a non-negative real number less than 1 (0 ≤ u ≤ 1), and v is a complex number such that u^2 + |v|^2 = 1, the effect of single-photon loss is to transform this state into

[0035]

[0036] This is also a valid logical state, and therefore it will cause undetectable errors in the system.

[0037] One potential way to circumvent this problem is to use cat states with a large number of coherent components.

[0038] Figure 1b An example embodiment is shown in which the quantum system carries cat states with four coherent states instead of two. For ease of reference later, let us refer to these four coherent cat states as "four-legged cats," or simply 4-cats, and the cat states with two coherent components as "two-legged cats," or simply 2-cats. This approach faces two distinct challenges. First, stabilizing these 4-cat states appears to be more challenging than stabilizing 2-cat states. Second, while 4-cat encoding can theoretically distinguish individual errors, doing so requires monitoring much more frequently than the timeframe corresponding to the boson loss probability, which may be quasi-continuous. For example, in one embodiment, a missed single error could lead to a second error causing a phase-flipping error, thus potentially becoming undetectable.

[0039] It is important to note that the second problem depends to some extent on the fact that 4-cat logic encoding has rotational symmetry. Therefore, in some embodiments, an N-cat logic encoding scheme that does not have rotational symmetry can be preferably chosen, i.e., N-cat logic encoding, where any loss of N photons (where n < N) results in a state outside the logic basis, and is therefore detectable and potentially correctable.

[0040] This is particularly relevant to the case of 3-cat logic encoding, as described below, which appears to be particularly promising, with illustrative implementations such as... Figure 1c As shown. The first logic state can be associated with two of the coherent components, while the second logic state can be associated with other coherent components (or linear combinations thereof).

[0041] Research has found that rotational asymmetry conditions can also be satisfied in other N-cat logic encodings where N is greater than 2 (including when N = 4). For example, when N is also a prime number, the non-rotational symmetry condition necessarily exists, but it may also necessarily exist in other relatively simple schemes, some of which will be described in detail below. In other words, the following specification will explore encoding schemes in physical systems that can represent a richer set of N (where N > 2) different quantum states, and the chosen configurations do not exhibit rotational symmetry. Logical qubits can be defined in this way. Furthermore, the following specification will explore techniques that allow operations to be performed on such qubits, which can correct for errors that may occur due to unwanted interactions with the environment, resulting in photon loss.

[0042] Figure 1 shows the phase space representations of a) 2-cat state, b) 4-cat state, and c) 3-cat state, illustrating the superposition of coherent states (great circles) and the interference fringes they produce.

[0043] A common weakness of 2-cat states and at least some higher-order coding schemes can perhaps be most generally summarized in the following configurations, where the cat states encoded by the logical states cause the resulting state change when a boson loss occurs to lead to a logical subspace indistinguishable from the initial logical subspace. These configurations may not allow for the simple detection of boson loss by determining the state. A general technique for addressing this problem is to use a configuration where one (or ideally one or more) boson loss of the initial state leads to a logical subspace significantly different from the initial logical subspace.

[0044] One example of implementing this general technique is choosing an encoding scheme that does not possess rotational symmetry. In fact, in these configurations, the boson loss of the initial state can lead to a logical subspace that is significantly different from the initial logical subspace, allowing for detection errors. A partitioning scheme can be used to divide the logical subspace into two logical states. Several example partitioning schemes that do not possess rotational symmetry are given below, such as partitioning schemes using a prime number of coherent components or partitioning schemes that encode one logical state in one coherent component and another logical state in a combination of the other coherent components. The simplest case might be the cat encoding scheme with three coherent states, where the first logical state is encoded in the first coherent state, and the second logical state is encoded in a combination of the other two coherent states. This simplest case may be preferred in some embodiments for various reasons, such as further promoting stabilization, but it should be understood that in other partitioning schemes, the boson loss leads to different logical substates and is therefore detectable. For example, it should be noted that some suitable alternative encoding schemes may arise from the careful selection of linear combinations chosen to define logical states that break the symmetry of the coherent component configuration in the cat state that is otherwise symmetric.

[0045] Before exploring further example implementations, we will first explore a first example based on a 3-cat configuration.

[0046] Example Implementation

[0047] Attribution of coherent states to logical states

[0048] Now, we will describe in detail an example embodiment based on a quantum subsystem (where N=3, and it may be called a quantum ternary (qutrit) or cubic cat state), which is partitioned by selecting a single coherent state as logic |0> and selecting a combination of two other coherent states as logic |1> (and vice versa).

[0049] We begin with the task of defining the states of two logical qubits. For N=3, there are 3 naturally coherent states, namely...

[0050] |α>,|αe 2πi / 3 >,|αe -2πi / 3 >

[0051] Where α is a positive real number, often referred to as the "size" of the cat, because in the phase space representation, it corresponds to the radius of the circle containing each coherent component state (often called the "leg") of the cat's states. From these three coherent states, we select an effective partition, where the first of these three states represents the |0> logic state, and the other two states represent the |1> logic state. Then, the logic |0> is simply defined as...

[0052]

[0053] For the |1> logical state, a selection must be made to define it. In this example, for simplicity, we made the following selection:

[0054]

[0055] The '~' symbol indicates that the state is proportional to the right-hand side of the above expression, because the definition of the state includes an additional normalization factor. For fairly large values ​​of α, this normalization is typically a number close to 1.

[0056] We define a third quasi-orthogonal state, which proves useful in further demonstrations, namely...

[0057]

[0058] We call this state the "ancilla" state.

[0059] Starting from the general logic qubit state

[0060]

[0061] in,

[0062]

[0063] Then, for single-photon or two-photon loss, the state will change in the following way:

[0064]

[0065]

[0066] If the system suffers a loss of 3 photons, the state in this model will be completely restored to its original state. This means that for any number of photon losses, there are only 3 states through which the system can cycle. Crucially, the two other states are not logically based because they involve a third quantum ternary state |2>. Therefore, they are distinguishable, and they do not corrupt the relative values ​​of u, v, and Φ, which carry the logical quantum information that must be preserved. This is achieved through the following process: 1) finding two subsets that appropriately partition the naturally coherent state, avoiding rotational symmetry in the phase space representation; and 2) defining two nontrivial and quasi-orthogonal or preferably orthogonal logical states on these subsets.

[0067] Stabilization of coherent states

[0068] To ensure a longer lifetime for the quantum information encoded in qubits, a stabilization process can be appropriately implemented. We propose a specific method to generate a Hamiltonian with the desired properties to stabilize the three-legged cat. The Hamiltonian in the proposed rotating frame is given by the following equation:

[0069]

[0070] Where K is the amplitude of the Kerr nonlinear term, ε3 is the amplitude of the three-photon actuator, and Δ is the detuning between the actuator and resonator frequencies. Careful selection of these parameters allows for the generation of a primary potential that stabilizes a cubic cat state with quasi-degeneracy and three coherent legs. Additional single-photon and two-photon actuators can be used to fine-tune the relative energy minimum of each state.

[0071] The logic behind this proposition is that it allows the use of naturally occurring Kerr terms, as well as 3-photon pumps and detuning terms.

[0072] To understand how it works, it is useful to rewrite the Hamiltonian in the displaced frame to see how appropriate choices of parameters allow certain terms to become negligible, thus making vacuum an eigenstate in the frame:

[0073]

[0074] It is understandable that there are additional Hermitian complements in the first three terms (omitted for clarity). A simple check reveals... The sole term in the array cannot be zero. However, this is only true if the parameter is chosen to ensure that it is relative to the sum of its parts. Keeping the detuned term small is not a problem. For example, this should be followed when choosing parameters such that |ε3|<<|Δ|, as shown below.

[0075] Returning to the first item, we wrote down

[0076] α=|α|e iθ ;

[0077] Then we get

[0078]

[0079] In this situation, the first term can be rewritten as

[0080]

[0081] This is quadratic in |α|, and allows real roots when the following determinant holds.

[0082]

[0083] In particular, when the following holds true,

[0084] |ε3|<<|△|

[0085] We get

[0086]

[0087] When the following holds true The compression terms in the code can be suppressed.

[0088]

[0089] Furthermore, if the condition for the absolute value of α (as described above) is satisfied, then The detuned term in the equation is approximately equal to -Δ, thus ensuring that when |ε3| << |Δ|, this term is significantly greater than a. 3 That is the purpose of this demonstration.

[0090] This proposed Hamiltonian stabilization is in Figure 2 The elemental potential is given in Chinese. Note that the elemental potential remains quasi-circular near its minimum, which helps to avoid compressive deformation in the steady state.

[0091] Operations on logical qubits

[0092] The operations performed on the proposed cubic cat qubits depend to some extent on the choice of the overall processor architecture. Many competing architectures exist. Example embodiments of two of these architectures are provided below: a quantum annealer and a gate-based quantum processor.

[0093] Depending on the type of architecture and the type of quantum subsystem used as the basis for the logical states, the implementation details of a quantum processor can vary considerably from embodiment to embodiment. However, in general, many architectures and types will involve using two or more quantum subsystems to carry the logical states. Indeed, as Figure 3 As shown, a typical quantum processor requires at least two quantum subsystems that are interconnected so that their coherent states interact in interactions typically involving entanglement. The type of quantum subsystem varies depending on the architecture. Most quantum systems implemented as boson-based quantum processors contain some form of resonator to carry the coherent states, and some form of driving hardware is used within the quantum subsystem to drive the coherent states, which can control multiple bosons within the quantum subsystem. The driving hardware is controlled by a component referred to herein as a controller, and this component is typically provided in the form of a classical computer. Quantum subsystems are typically cooled to very low temperatures and insulated from the environment. In quantum annealed architectures, quantum subsystems can be directly and operablely interconnected. In gate-based quantum computing, quantum subsystems are typically interconnected via couplers, which are used to selectively control the interactions between the quantum subsystems. The coupler is also a quantum subsystem operable to carry the coherent states of two or more connected quantum subsystems via their interactions, and the coupler is also driven by driving hardware, which, for convenience, can be controlled by the same controller. Now let’s look at an example of how to perform quantum computing using a 3-cat configuration.

[0094] A-Quantum annealing (QA)

[0095] In QA, qubits operate in a continuous manner from the start to the end of computation. The Hamiltonian applied to each physical qubit changes continuously from a simple initial Hamiltonian that is easy to prepare, and then gradually transforms into the final form representing the problem to be solved.

[0096] We can distinguish three important steps in this process: preparation (or initialization), evolution, and readout.

[0097] Initialization: Each individual physical qubit can be initialized independently, and in the simplest case, the parameters of each qubit can be identical. In most cases, the qubit is initialized in the |+> state (|+> = |0> + |1>). In the case of the proposed cubic cat encoding, this can be achieved by setting the cavity to a vacuum state and gradually increasing the effective parameters of the proposed Hamiltonian (i.e., the parameters of the driver). In cases where the initial state should differ from the simplest case described above, a more general initialization method can be used (described in the next section on gate-based models).

[0098] Evolution: In the evolution phase, the Hamiltonian gradually changes to represent the problem to be solved. Nontrivial problems involve entanglement between any given qubit and at least one other qubit. In most cases, the problem Hamiltonian involves only entanglement terms of the following form:

[0099]

[0100] That is, only σ z The sum of operator products. In this case, the inter-qubit interaction terms can be implemented in the Hamiltonian representing the problem Hamiltonian without error correction of the qubits. This is because σ z The operator only affects the complex phase of the 0 component relative to its orthogonal components, and the operation is the same for any of the three possible cyclic states of a given qubit at any given time.

[0101] Readout: The logical state of a qubit can be read out through a projection or quantum nondestructive measurement that distinguishes between coherent states with different phases by the interaction of a parametric beam splitter between a cubic transmon and a readout resonator.

[0102] B-Menji Model (GB)

[0103] In gate-based architectures, in principle, when performing logical operations involving one or more qubits, the qubits only need to operate in discrete time. In practice, however, in many proposed implementations, the qubits are operated quasi-continuously (or at least monitored) for error correction purposes.

[0104] In one of the proposed embodiments, we might be misled by the need for any continuous operation or monitoring of the qubits. When performing certain types of logic operations on one or more qubits, error correction operations may only be needed at discrete times. Of course, this method does not preclude continuous monitoring for other purposes when necessary.

[0105] Similar to the QA architecture, we also distinguish the three important steps in the computation process in the GB architecture: preparation (or initialization), operation cleanup (error elimination), and readout.

[0106] Operation cleanup: We start with the second step of the process description because initialization and reading are essentially sub-steps of this process.

[0107] When performing operations that cannot be described by the product of the Z operators, a step we call "cleaning" in this paper may be necessary because the later operator may not be affected by photon loss.

[0108] For other operations, restore the original state. This can likely only be achieved by using an additional physical source (i.e., an auxiliary source), the simplest of which is a single physical 2-cat qubit. The main idea is as follows:

[0109] 1) Apply entanglement operations to the quantum ternary / auxiliary system to entangle the two elements and embed the logical qubits into them;

[0110] 2) Reset the quantum ternary array to allow the logic qubits to be fully transferred to the auxiliary array;

[0111] 3) Quickly apply a reverse spin to bring the logical qubit back into the quantum ternary array, protecting it from the destructive effects of photon loss within the quantum ternary array; and

[0112] 4) Reset assistance.

[0113] The key element in the entanglement operation in step 1) above is implemented as a controlled phase gate for modification, with the goal of assistance and controlled by a cubic cat. The next section provides a description of an example sequence of the operational steps involved in this process.

[0114] Initialization: Initialization is performed by first preparing the required logical state in the auxiliary qubit, and then performing steps 3) and 4 as described above (in the operation cleanup).

[0115] Readout: Readout is performed by applying steps 1) and 2) as described above (in the operation cleanup), and then readout is performed on the auxiliary qubit.

[0116] Advanced instructions for performing cleanup operations.

[0117] Based on the general principles outlined above, the cleaning operation can be specifically applied to different embodiments. We will now explore the possible applicability to specific embodiments of the 3-cat described above, in order to provide a detailed example embodiment.

[0118] Based on the above discussion, the objective can be limited to recovery from single-photon or two-photon losses, since three-photon losses would restore the system to its original state. Therefore, the main error path cycles between three distinct states that preserve the embedded qubit information, eliminating the need for continuous monitoring of photon losses and allowing recovery only when necessary, i.e., when a non-preserved gate is applied.

[0119] Based on the general principles outlined above, the cleaning operation can be specifically applied to different embodiments. We will now explore the possible applicability to specific embodiments of the 3-cat described above, in order to provide a detailed example embodiment.

[0120] exist Figure 4 In order to represent the tensor product of the auxiliary qubit and the quantum ternary, the cartoon Wigner diagram of the two legs of the auxiliary 2-cat qubit is represented as two large red circles, and the corresponding cartoon Wigner diagram of the quantum ternary is represented in each of these circles.

[0121] Before correction, the quantum ternary is in state Ψ, which has only three possible states, i.e.

[0122]

[0123] in,

[0124]

[0125] And as above

[0126]

[0127] and

[0128]

[0129] One approach is to simply clone the state of the quantum ternary, measure one of the three possible states, and appropriately correct the original quantum ternary. Unfortunately, the no-cloning theorem forbids this. However, it does not forbid the movement of quantum information from one subsystem to another. While simply moving around the vicinity of the entire quantum ternary makes little sense, logical qubits can be moved from within the physical quantum ternary to an additional physical qubit (called an auxiliary or supplementary qubit) coupled to the physical quantum ternary, leaving only the unwanted dephase, and then the original quantum ternary is reset to eliminate the unwanted dephase caused by noise (see [link to original text]). Figure 4 ).

[0130] The first operational step is to prepare the auxiliary qubit in the |+> state by rotating πi / 2x, while still decoupling it from the quantum ternary ( Figure 4 (Lines 2 and 3). After this step, the combined qubit-qubit ternary system is in the following state:

[0131] |+> b |ψ> t ,

[0132] The subscripts b and t represent qubit and quantum ternary, respectively.

[0133] The next step, and the main step in the process, is to entangle the quantum ternary with the auxiliary quantum bit through a phase gate CZ = ZZ(π / 2) controlled by a specific quantum bit-quantum ternary. CZ = ZZ(π / 2) is analytically described as follows:

[0134]

[0135] Transform the state of the combination to:

[0136]

[0137] The final step is as follows: Figure 4 As shown in lines 3 and 4.

[0138] In the specific case where the auxiliary qubit is a two-legged cat state, we can achieve this operation by implementing the following auxiliary / quantum ternary interaction Hamiltonian:

[0139]

[0140] Among them, J b,t It is the coupling factor for the beam splitter interaction between the auxiliary and quantum ternary components. This allows us to realize the desired modified controlled phase operator.

[0141]

[0142] in,

[0143] θ = 3J b，t α 2 t gate ,

[0144] Among them, t gate It is the duration of the interaction.

[0145] The final step involves resetting the quantum ternary to its |0> state, thanks to the previous -π / 2x rotation of the auxiliary qubit ( Figure 4 (Lines 4 to 5 and lines 5 to 6).

[0146] Patterns and Interactions

[0147] Figure 5 The diagram illustrates the modes and interactions in an example embodiment where a 2-cat quantum system is attached to each 3-cat subsystem for cleanup purposes as described above. In the photonic implementation, the mode carrying the three-legged cat can interact with a time-controlled beamsplitter that interacts with an auxiliary mode carrying the two-legged cat state (2-cat). The state of the two-legged cat can be read out to a readout mode through interaction with the time-controlled beamsplitter.

[0148] Example circuit

[0149] Figure 6 An example circuit for implementing the above process is shown. Three-legged cat and two-legged cat states exist in two cubic transporters, each consisting of a parallel small Josephson junction and N ≥ 2 larger junctions, along with a large shunt capacitance. A finite magnetic flux is applied in the junction loop to achieve second-order nonlinearity. The magnetic flux can be applied through individual flux lines (as shown) or through a single flux line. Individual drive ports allow for pumping of the cubic transporters using single-photon and three-photon pumps (one-photon pump and two-photon pump) to stabilize the 3-cat (2-cat) state. The two cubic transporters are capacitively coupled, resulting in a fixed beamsplitter interaction. When the two modes are far detuned to each other, the beamsplitter interaction can lead to a static and undesirable dispersed ZZ-type interaction. This interaction can be suppressed by additional continuous drive of either cubic transporter. The desired time-controlled beamsplitter interaction between the two cubic transporters is parameterized and activated by driving either mode at their frequency difference. Quantum nondestructive readout of the two-legged cat state is performed by driving the cubic transporter at the frequency difference between the cubic transporter and the resonator, thereby parameterizing and activating the beam splitter interaction between the corresponding cubic transporter and the readout resonator.

[0150] In one example, the circuit may be equipped with the following drivers:

[0151] 3-Cat Stabilization: Single-Photon Driver at Cubic Transmitter Frequency

[0152] Stabilization of the 3-Cat: A Three-Photon Actuator at Three Times the Cubic Transmitter Frequency

[0153] 2-Cat Stabilization: Two-Photon Driver at Twice the Cubic Transmitter Frequency

[0154] Operation of 2-Cat: Single-Photon Driver at Cubic Transmitter Frequency

[0155] 2-Cat Readout: Parameterized Driver Based on the Difference Between the 2-Cat Cubic Transmitter and the Corresponding Resonator

[0156] Interaction between 3-cats and 2-cats: A parameterized driver for the frequency difference of cubic transporters in time-controlled ZZ-type interactions.

[0157] Interactions between 3-cats and 2-cats: Parameterized drivers for suppressing static ZZ-type interactions

[0158] Example Alternative Embodiments

[0159] Bosons other than photons

[0160] The proposed scheme can be implemented in alternative architectures where the fundamental excitations are modes of other types of bosons. The most notable candidate is a phonon-based architecture. A significant advantage of this architecture is the demonstration of a phonon resonator with a lifetime of up to 1 second. Another possible implementation is using a magnetostatic mode, where the magnon is a boson excitation. A major challenge in the latter architecture is its short lifetime of approximately several hundred nanoseconds. In both alternative physical implementations, the nonlinearity required in the proposed scheme is likely achieved by coupling the boson modes to a nonlinear superconducting circuit.

[0161] The likelihood of N>3-legged cat stabilization

[0162] The second-order linearity (Pockels) and third-order nonlinearity (Kerr) present in the cubic transporter provide the necessary terms for potentially engineering N-photon actuators with N>3. However, it will be noted that the efficiency of such actuators may decrease as N increases.

[0163] Figures 7a to 7j This is a collection of schematic diagrams representing examples of encodings that can form the basis of qubits. In the general form, it is suggested that encoding be performed in the context of N-legged cat (or N-cat) states (where N>2). When represented in the phase space of the resonator cavity, these N individual states are equidistant on the circumference.

[0164] The N individual states are divided into two subsets, represented by either white or black dots. Each subset represents one of the two logical states of a qubit. The logical states are configured to be rotationally asymmetric, meaning there is no angle smaller than a full rotation, through which the logical state can be rotated and returned to the same configuration. Figures 7a to 7f A method is presented to achieve this rotational asymmetry based on the state belonging to a given subset, which will be referred to as asymmetry due to the subset configuration. Figure 7j The following example is given, where rotational asymmetry is caused by the zero coefficient belonging to a subset configuration (otherwise rotationally symmetric), and can be referred to as asymmetry due to coefficient configuration. Let's first discuss rotational asymmetry based on subset configuration.

[0165] For example, if N is a prime number greater than 2, then this condition will be automatically satisfied in all possible partitioning configurations between the two subsets. However, it should be noted that N is odd and rotational symmetry is not guaranteed to be absent, such as... Figure 7i As shown. Figure 7g and Figure 7h It also indicates the existence of a rotationally symmetric partition configuration. Figures 7a to 7f Various examples of non-rotational symmetry are given.

[0166] Various methods or rules can be implemented to facilitate or automatically achieve partitions that do not exhibit rotational symmetry. For example, regardless of whether N is a prime number, partitions can be achieved by selecting, for instance, ... Figures 7a to 7c The two contiguous subsets shown achieve a partitioning configuration without rotational symmetry. By selecting a single state to represent one of the two logical states and selecting the remaining N-1 states to represent the other logical state, a simpler partitioning can always be obtained; this is... Figure 7c and Figure 7f The configuration shown is an example. However, for relatively large N, the latter configuration choice makes achieving near orthogonality between the two logical states more challenging, which is an additional requirement discussed below. In the case of odd N, another suitable partitioning configuration can be easily obtained by alternating partitions, such as... Figure 7d As shown.

[0167] Generally, for simplicity, a smaller value of N is preferred, especially in cases where stabilization issues may exist. For example, N could preferably be chosen as a value of 3, 4, or 5. Such as Figure 7f As shown, N=3 may be more preferred in some embodiments and is chosen as the basis for providing the detailed example embodiments given below.

[0168] Once appropriate subsets are chosen, each quantum state (|0> and |1>) can be defined by any nontrivial linear combination of the states that make up their respective subsets, where linear combinations allow for complex coefficients, and the chosen states can be appropriately normalized. Thus, if not exactly orthogonal to two logical states, we should at least ensure approximate orthogonality between them.

[0169] The encoding of the logic state can also be adjusted in a rotationally symmetric partitioning configuration; however, the specific choice of coefficients in a linear combination breaks the symmetry. A simple example of this variant is... Figure 7j As shown, the states marked with the "X" symbol are assigned coefficients of zero, effectively transforming this 6-cat rotationally symmetric partitioning configuration into a 3-cat asymmetric configuration. These two concepts can be combined, and the asymmetric configuration can be further assigned one or more zero coefficients while maintaining non-rotational symmetry.

[0170] It should be noted that other embodiments may exist. This concept can be referenced. Figures 8a to 8e To elaborate further, Figures 8a to 8e This schematically represents the phase shift from the initial state. Figure 8a In the phase space representation of the graph, two distinct logic states are defined as i) a superposition of white dot groups and ii) a superposition of black dot groups. The definition of each logic state involves assigning complex numbers, namely amplitude and phase (which can be zero, in which case the individual coherent states are excluded from the logic state definition). It is well known that the sum of the squares of the complex amplitudes of each logic state equals 1, and the dot product of two logic states is minimized. The initial state is a given superposition of logic states representing logical information.

[0171] exist Figure 8b In the initial state, the phase of each coherent state is represented by a pointer on a clock. Phase shifts may occur in certain types of unintended or unexpected interactions with the environment (i.e., interactions that do not correspond to interactions driven by controlled commands). Different individual coherent states experience different phase shifts. For example, Figure 8c This represents the situation where the phase shift does not affect the first coherent state (on the right-hand side) and the phase change is proportional and increasingly larger, which is represented by the position of the clock pointer changing in a counterclockwise circular sequence from the first coherent state to the next coherent state.

[0172] As this example illustrates, if the different coherent states associated with each logical state remain in phase with each other, but out of phase with the coherent state associated with another logical state, logical information is lost. This is undesirable, but can be avoided by defining logical states in a way that prevents this from happening. Many different techniques can be used to ensure this does not occur, many of which have been proposed above to form the basis of numerous examples.

[0173] However, other techniques can also be used in other embodiments. For example, let us consider the following case where S is the set of all n coherent states of the system, A and B are strict subsets of S, i.e., A and B are strictly contained in S (but A is not equal to N, and B is not equal to N), where the first logical state consists of elements of A (which may include all elements of A or any subset thereof), and where the second logical state consists of elements of B (which may include all elements of B or any subset thereof). In this case, if A and B are chosen such that the following i) and ii) are satisfied, an uncorrectable error may occur:

[0174] i) Both A and B are rotationally symmetric, regularly spaced subsamples of set S (e.g., states 1 and 0 in the upper row, and state 0 in the lower row). and

[0175] ii) B can be obtained by rotating A (B is simply a rotated version of A, and vice versa).

[0176] Therefore, to avoid potential problems, we can choose A and B, such that:

[0177] i) At least one logical state is not a subset of a rotation-symmetric subset of S. or

[0178] ii) If both of them are, then they should not be subsets of two rotationally symmetric subsets of S that are identical before the rotation.

[0179] and Figures 8a to 8e Compared to the situation in the middle, Figures 9a to 9f The same representation logic was used to present another case. Figures 9a to 9f In this case, the logical state is defined as follows: from the initial state ( Figure 9b ) Figure 9c , Figure 9d , Figure 9e and Figure 9f Any possible increment of the phase shift shown is limited to obtaining one or more modified states located outside the logical subspace that simultaneously preserve logical information (e.g., Figure 9c , Figure 9d , Figure 9e , Figure 9f ), or return to the initial state. In fact, in Figures 9c to 9f In each intermediate phase shift shown, the coherent states of each logic state are in different phases between them, and out of phase with the coherent states associated with other logic states.

[0180] The definition of a logical state can involve assigning different combinations of complex numbers to each coherent state. Figures 7a to 7i , Figure 8a and Figure 9a In the cases shown, complex numbers are weighted to either 1 or zero, and combinations of coherent states that are weighted to 1 in the definition of the first logical state are weighted to zero in the second logical state, and vice versa. Figure 7j In the case shown, some coherent states are weighted to zero in the definition of the two logical states and are therefore not used. In different embodiments, the exact choice of the combination of complex numbers associated with each coherent state and defining the two logical states can vary; the logical states can share some or all of the coherent states and can simply be weighted differently, including values ​​other than 1, to minimize their dot product. Similarly, in Figure 9b In this context, all coherent states are synchronized in the initial state, but it is understandable that coherent states may be out of sync with each other in the initial state.

[0181] It should be understood that the expressions "computer," "classical computer," or "controller" as used herein should not be interpreted in a restrictive manner. "Computer" in a broad sense generally refers to a combination of one or more processing units of some form and some form of storage system accessible to the processing units. "Controller" in a broad sense generally refers to a device that performs control functions and can be a computer or other type of device. The memory system of a computer can be of a non-transitory type. The singular form of "computer" as used herein includes, within its scope, a combination of two or more computers working together to perform a given function, regardless of whether these two or more computers are local, remote, or distributed. Furthermore, the expression "computer" as used herein includes, within its scope, a portion of the capabilities of a given processing unit.

[0182] The processing unit can be implemented in the form of a general-purpose microprocessor or microcontroller, a digital signal processing (DSP) processor, an integrated circuit, a field-programmable gate array (FPGA), a reconfigurable processor, a programmable read-only memory (PROM), etc.

[0183] Memory systems can include suitable combinations of any appropriate types of computer-readable storage, which may be internal or external and may be directly accessed by a processor via wired or wireless means or via a network such as the Internet. Computer-readable storage can be implemented in the form of random access memory (RAM), read-only memory (ROM), optical disc read-only memory (CDROM), electro-optical memory, magneto-optical memory, erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), ferroelectric RAM (FRAM), etc.

[0184] A computer may have one or more input / output (I / O) interfaces to allow communication with a human user and / or another computer via associated input, output, or input / output devices (such as a keyboard, mouse, touchscreen, antenna, port, etc.). Each I / O interface enables the computer to communicate and / or exchange data with other components, access and connect to network resources, serve applications, and / or perform other computing applications by connecting to a network (or multiple networks) capable of carrying data. These networks include the Internet, Ethernet, Common Old-Style Telephone Service (POTS) lines, Public Switched Telephone Network (PSTN), Integrated Services Digital Network (ISDN), Digital Subscriber Line (DSL), coaxial cable, fiber optic, satellite, mobile devices, wireless (e.g., Wi-Fi, Bluetooth, WiMAX), SS7 signaling networks, fixed lines, local area networks (LANs), wide area networks (WANs), etc.

[0185] It should be understood that a computer can perform functions or processes through hardware or a combination of hardware and software. For example, hardware may include logic gates included as part of a silicon chip that is a processor. Software (e.g., application programs, processes) may be in the form of data, such as computer-readable instructions stored in non-volatile computer-readable memory accessible by one or more processing units. With respect to a computer or processing unit, the statement "configured as" refers to the presence of hardware or a combination of hardware and software operable to perform the associated functions.

[0186] It is understood that the examples described and illustrated above are merely exemplary. The appended claims indicate the scope.< / n>

Claims

1. A quantum system comprising: A quantum subsystem that can be driven to carry a boson cat state with N coherent states, where N is greater than 2, the N coherent states defining a computational space; Driving hardware, which is operatively connected to drive the quantum subsystem to carry the boson cat state; as well as A controller, operably connected to control the drive hardware according to control commands. The at least one type of interaction between the boson cat state and the environment, which is not derived from control commands, causes a phase shift in the N coherent states. The control instructions include a function that generates an initial state of the boson cat state, the initial state being defined as a given superposition of two distinct logical states representing logical information, the two distinct logical states spanning a logical subspace within the computational space, wherein any phase shift from the initial state results in i) The modification status of the logical information is stored outside the logical subspace, or ii) Return to the initial state. Where N=3, The quantum subsystem is a resonator with a resonator frequency, the driving hardware includes a three-photon driver with a driver frequency, and the cat state is based on the Hamiltonian. Driven, where K is the amplitude of the Kerr nonlinear term, ε3 is the amplitude of the three-photon driver, and Δ is the detuning between the driver frequency and the resonator frequency, and The controller is configured to control the driver hardware, such that... ,and .

2. The quantum system according to claim 1, wherein, The N coherent states can be represented as points around a circle in the phase space of the quantum subsystem, driving the quantum subsystem includes controlling a plurality of bosons in the quantum subsystem, and wherein the logic state is rotationally asymmetric when the boson cat state rotates around the circle.

3. The quantum system according to claim 1, comprising at least two quantum subsystems, each quantum subsystem having a corresponding driving hardware, both of the at least two driving hardwares being controlled by the controller, the quantum subsystems being interconnected to allow the coherent states of the interconnected quantum subsystems to interact.

4. The quantum system according to claim 3, wherein, Each quantum subsystem in the quantum subsystem is connected to other quantum subsystems directly or via other quantum subsystems in the quantum subsystem.

5. The quantum system according to claim 3, wherein, Each of the quantum subsystems is connected to at least one other quantum subsystem in the quantum subsystem via a coupler, the coupler being drivable to selectively allow or prevent interaction between the coherent states of the connected quantum subsystems.

6. The quantum system according to claim 1, wherein, The quantum subsystem can be driven to carry coherent states. The first logical state in the logical states is encoded as the first coherent state in the coherent states, and the second logical state in the logical states is encoded as the other two coherent states.

7. The quantum system of claim 1, further comprising an auxiliary subsystem, the auxiliary subsystem being driveable to carry a cat state having at least two coherent states, the auxiliary subsystem being selectively connected such that the coherent states of the auxiliary subsystem are selectively entangled or de-entangled with the coherent states of the quantum subsystem.

8. The quantum system according to claim 7, wherein, The auxiliary subsystem can be driven to carry a cat state with two coherent states.

9. The quantum system according to claim 7, wherein, The boson is a photon, the quantum subsystem includes a first cubic transporter having a first transporter frequency, the auxiliary subsystem includes a second cubic transporter having a second transporter frequency and a readout resonator coupled to the second cubic transporter and having a resonant frequency, the driving hardware includes a single-photon driver connected to the first cubic transporter and operating at the first transporter frequency, a three-photon driver connected to the first cubic transporter and operating at three times the first transporter frequency, a two-photon driver connected to the second cubic transporter and operating at twice the first transporter frequency, a single-photon driver connected to the second cubic transporter and operating at the second transporter frequency, a first parameterized driver operating at the difference in transporter frequencies for connecting time-controlled ZZ-type interactions between the first cubic transporter and the second cubic transporter, a second parameterized driver connected between the first cubic transporter and the second cubic transporter and configured to suppress static ZZ-type interactions, and a third parameterized driver operating at the difference between the second cubic transporter and the corresponding resonator.

10. The quantum system according to claim 1, wherein, The first logic state is encoded in the first coherent state of the coherent state, and the second logic state is encoded in the other coherent states of the coherent state.

11. The quantum system according to claim 1, wherein, Rotational symmetry is broken by assigning a coefficient of zero to at least one of the coherent states.

12. The quantum system according to claim 1, wherein, The bosons mentioned are photons.

13. A method for carrying a boson cat state with N coherent states in a quantum subsystem, wherein N is greater than 2, and the N coherent states define a computational space, the method comprising: An initial state is generated for the boson cat state, which is a given superposition of two distinct logical states representing logical information. These two distinct logical states span a logical subspace within the computational space. The two logical states are defined such that any phase shift of the boson cat state from the initial state, originating from an uncontrolled interaction between the boson cat state and its environment, is limited to either resulting in one or more modified states located outside the logical subspace that simultaneously retain the logical information, or a return to the initial state. The generation of the initial state includes setting the cavity of the quantum subsystem to a vacuum state and gradually increasing the driver parameters of the Hamiltonian. , making ,and .

14. The method according to claim 13, wherein, The N coherent states can be represented as points around a circle in the phase space of the quantum subsystem. The generation includes controlling multiple bosons in the quantum subsystem, and the logic state is rotationally asymmetric as the boson cat states rotate around the circle due to boson losses of the quantum subsystem that occur outside of controlling the multiple bosons.

15. The method of claim 13, wherein the quantum subsystem is a first quantum subsystem, the method further comprising driving a cat state having N coherent states in a second quantum subsystem, and causing the coherent states of the first quantum subsystem to interact with the coherent states of the second quantum subsystem.

16. The method of claim 13, further comprising detecting logic errors in the coherent state, including detecting the modified state.

17. The method of claim 16, further comprising driving a cat state having at least two coherent states in an auxiliary subsystem, moving the logic state from the cat state of the quantum subsystem to the cat state of the auxiliary subsystem, resetting the coherent state of the quantum subsystem, and returning the logic state from the cat state of the auxiliary subsystem to the cat state of the quantum subsystem.

18. The method according to claim 17, wherein, The cat state of the quantum subsystem is carried in a first photon mode, and the cat state of the auxiliary subsystem is carried in a second photon mode.

19. The method according to claim 18, wherein, The interaction is performed through the interaction of time-controlled beam splitters.

20. The method of claim 18, further comprising using time-controlled beam splitter interactions to read the cat state of the auxiliary subsystem.

21. The method of claim 13, further comprising generating a cat state having at least two coherent states in an auxiliary subsystem, applying an entanglement operation between the quantum subsystem and the auxiliary subsystem such that the logic state is embedded in both, resetting the quantum subsystem such that the logic state is carried solely by the auxiliary subsystem, applying a reverse rotation to bring the logic state back from the auxiliary subsystem to the quantum subsystem, and resetting the auxiliary subsystem.

22. The method of claim 13, further comprising generating a cat state having at least two coherent states in an auxiliary subsystem, wherein encoding two distinct logic states includes preparing the two distinct logic states in the auxiliary subsystem, and applying a rotation to bring the logic states back from the auxiliary subsystem to the quantum subsystem.

23. The method of claim 15, further comprising driving a cat state having at least two coherent states in an auxiliary subsystem, applying an entanglement operation between the quantum subsystem and the auxiliary subsystem such that the logic state is embedded in both, resetting the quantum subsystem such that the logic state is carried solely by the auxiliary subsystem, and performing a readout on the auxiliary subsystem.

24. The method of claim 21 or 23, wherein the applied entanglement operation is implemented such that the auxiliary subsystem is set as a target and the quantum subsystem is set as a controlled modified phase gate.

25. A non-transitory computer-readable storage medium storing control instructions, the control instructions comprising: When executed by a computer, a function is used to generate the initial state of a boson cat state having N coherent states in a quantum subsystem, where N is greater than 2, the N coherent states defining a computational space, the initial state being defined as a given superposition of two distinct logical states representing logical information, the two distinct logical states spanning a logical subspace within the computational space, and wherein at least one type of interaction between the boson cat state and the environment, not originating from the control command, causes a phase shift in the N coherent states. The definitions of the two distinct logical states restrict the boson cat state from evolving into a modified state located outside the logical subspace while preserving the logical information, or returning to the initial state, at any phase shift from the initial state. The generation of the initial state includes setting the cavity of the quantum subsystem to a vacuum state and gradually increasing the driver parameters of the Hamiltonian. , making ,and .

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