A system, method, and control instruction for simultaneously preventing bit-flipping errors and phase-flipping errors in a quantum subsystem.

Abstract

The system may include a quantum subsystem drivable to provide a bosonic cat state having N coherent states, where N is greater than 2, the N coherent states defining a computational space, and a controller operatively connected to control the driving hardware in accordance with control instructions, where at least one type of interaction between the bosonic cat state and the environment not resulting from the control instructions results in a phase shift of the N coherent states, the control instructions including functionality for generating an initial state of the bosonic cat state 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, where any phase shift from the initial state results in a modified state outside the logical subspace while preserving the logical information, or returning to the initial state.

Description

[Technical Field]

[0001] The present invention relates to a system, method, and control instructions for simultaneously preventing bit inversion errors and phase inversion errors in a quantum subsystem. [Background technology]

[0002] Recently, the field of quantum computing has become an active area of ​​research because it has the potential to disrupt modern computing, communications, and cryptography.

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

[0004] One of the main challenges faced when encoding quantum information in actual physical devices is balancing two conflicting requirements. Firstly, quantum information is inherently fragile and susceptible to destructive causes of decoherence. Unfortunately, most interactions with the environment cause decoherence. Therefore, systems designed to store and process quantum information are typically isolated as much as possible from interactions with the environment. Secondly, for quantum information to be useful, it must be processed and ultimately accessible, which means that the quantum information must be controllable through interactions with an external control system, and for the processor to be efficient, this control of interactions must be performed at a relatively high frequency.

[0005] Due to this second requirement, avoiding errors in quantum information processing is extremely difficult. One general-purpose technique stemming from the field of classical information processing attempts to partially correct these errors through redundancy, such as redundantly encoding quantum information on additional physical resources. This technique is costly, and in practical applications, the typical overhead can be around 10,000 times, making it unattractive for many real-world applications.

[0006] A more recent approach involves encoding qubits in a richer quantum subsystem so that quantum information is redundantly encoded within a single physical element. A promising method of this approach is bosonic coding, where the phase space of each system can provide a theoretically infinite-dimensional Hilbert space for encoding information.

[0007] In an infinitely sized coding space, there are theoretically infinitely many methods for coding qubits. In practice, however, a few promising candidates with interesting properties have recently emerged. One such candidate is the so-called CAT state coding, where qubits are coded by superimposing two quasi-classical coherent states of the system. Recent studies on CAT state coding have yielded interesting results, but further improvements are still needed. Specifically, it is still necessary to block both bit-flipping errors and phase-flipping errors simultaneously. [Overview of the project] [Means for solving the problem]

[0008] In one embodiment, a method is provided for hosting a cat state of a boson having N coherent states, where N is greater than 2, and the N coherent states define a computation space. The method includes the step of generating an initial state of the cat state of a boson, which is a given superposition of two distinct logical states representing logical information, the two distinct logical states spread across a logical subspace in the computation space, and the two logical states are defined such that any phase shift of the cat state of a boson from the initial state resulting from an uncontrolled interaction between the cat state of the boson and the environment results in one or more modified states outside the logical subspace, while preserving the logical information, or returns to the initial state.

[0009] In another embodiment, a quantum system is provided, the quantum system comprising a quantum subsystem, driveable to provide a cat state of bosons having N coherent states, where N is greater than 2, and the N coherent states define a computation space; drive hardware operably connected to drive the quantum subsystem to provide the cat state of bosons; and a controller operably connected to control the drive hardware according to control instructions, wherein at least one type of interaction between the cat state of bosons and the environment, not arising from control instructions, results in a phase shift of the N coherent states, and the control instructions include a function to generate an initial state of the cat state of bosons, defined as a given superposition of two distinct logical states representing logical information, the two distinct logical states extending into a logical subspace within the computation space, and any phase shift from the initial state results in a modified state outside the logical subspace, while retaining the logical information, or returns to the initial state.

[0010] In another embodiment, a control instruction is provided which is stored in non-temporary computer-readable memory, and which, when executed by a computer, generates an initial state of a boson cat state having N coherent states in a quantum subsystem, wherein N is greater than 2, the N coherent states define a computation space, the initial state is defined as a given superposition of two distinct logical states representing logical information, the two distinct logical states extend into a logical subspace within the computation space, and at least one type of interaction between the boson cat state and the environment, not arising from the control instruction, results in a phase shift of the N coherent states; and the definition of the two distinct logical states, the definition constrains the expansion of the boson cat state from the initial state to a modified state outside the logical subspace, or back to the initial state, while retaining the logical information, depending on either of the phase shifts.

[0011] In another embodiment, a quantum system is provided, the quantum system being a quantum subsystem driveable to provide 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 subsystem, where N is greater than 2; a drive hardware operably connected to drive the quantum system to provide the cat state by controlling the number of bosons of the quantum system; and a controller operably connected to control the drive hardware to encode two distinct logical states having N coherent states, such that as the cat state rotates around the circle, the logical states become rotationally asymmetric due to the loss of bosons in the quantum system.

[0012] In another embodiment, a method is provided for performing quantum processing, which includes the step of driving a cat state having N coherent states in a quantum subsystem, where the N coherent states can be represented as points around a circle in the phase space of the quantum subsystem, where N is greater than 2, and the driving step includes the step of controlling the number of bosons of the quantum system and the step of encoding two distinct logical states having N coherent states, where as the cat state rotates around the circle, the logical states become rotationally asymmetric due to the loss of bosons in the quantum system.

[0013] Those skilled in the art will find many further features and combinations of features relating to the present improvement apparent upon reading this disclosure.

[0014] In the drawing, it is as follows: [Brief explanation of the drawing]

[0015] [Figure 1] This figure shows a) a 2-cat state, b) a 4-cat state, and c) a more detailed phase space representation of the 3-cat state, illustrating the superposition of coherent states (large circles) and the interference fringes generated by the coherent states. [Figure 2] This is a diagram of the metapotential representation of Hamiltonian stabilization. [Figure 3] This is a schematic diagram of a quantum system that can be implemented as a quantum processor. [Figure 4] This is a schematic diagram illustrating the cleaning process step by step. [Figure 5] This is a schematic diagram illustrating the modes of a logic qubit embedded in a three-legged cat state (3-cat), as well as the interactions that enable stabilization and quantum error correction. [Figure 6] This is a schematic diagram of an exemplary circuit for implementing the proposed protocol. [Figure 7a] In phase space, N states are a set of schematic diagrams representing N individual states that can be associated with logical states and form a basis for qubit coding. [Figure 7b] A set of schematic diagrams representing N individual states in a phase space, where the N states are associated with logical states and can form a basis for qubit encoding. [Figure 7c] A set of schematic diagrams representing N individual states in a phase space, where the N states are associated with logical states and can form a basis for qubit encoding. [Figure 7d] A set of schematic diagrams representing N individual states in a phase space, where the N states are associated with logical states and can form a basis for qubit encoding. [Figure 7e] A set of schematic diagrams representing N individual states in a phase space, where the N states are associated with logical states and can form a basis for qubit encoding. [Figure 7f] A set of schematic diagrams representing N individual states in a phase space, where the N states are associated with logical states and can form a basis for qubit encoding. [Figure 7g] A set of schematic diagrams representing N individual states in a phase space, where the N states are associated with logical states and can form a basis for qubit encoding. [Figure 7h] A set of schematic diagrams representing N individual states in a phase space, where the N states are associated with logical states and can form a basis for qubit encoding. [Figure 7i] A set of schematic diagrams representing N individual states in a phase space, where the N states are associated with logical states and can form a basis for qubit encoding. [Figure 7j] A set of schematic diagrams representing N individual states in a phase space, where the N states are associated with logical states and can form a basis for qubit encoding. [Figure 8a] A diagram of a schematic representation of the phase evolution due to successive boson losses for the definition of two exemplary logical states. [Figure 8b] A diagram of a schematic representation of the phase evolution due to successive boson losses for the definition of two exemplary logical states. [Figure 8c] This diagram provides a schematic representation of the phase expansion due to the loss of consecutive bosons for the definitions of two exemplary logical states. [Figure 8d] This diagram provides a schematic representation of the phase expansion due to the loss of consecutive bosons for the definitions of two exemplary logical states. [Figure 8e] This diagram provides a schematic representation of the phase expansion due to the loss of consecutive bosons for the definitions of two exemplary logical states. [Figure 9a] This diagram shows a schematic representation of the phase expansion due to the loss of consecutive bosons for the definitions of two exemplary logical states. [Figure 9b] This diagram shows a schematic representation of the phase expansion due to the loss of consecutive bosons for the definitions of two exemplary logical states. [Figure 9c] This diagram shows a schematic representation of the phase expansion due to the loss of consecutive bosons for the definitions of two exemplary logical states. [Figure 9d] This diagram shows a schematic representation of the phase expansion due to the loss of consecutive bosons for the definitions of two exemplary logical states. [Figure 9e] This diagram shows a schematic representation of the phase expansion due to the loss of consecutive bosons for the definitions of two exemplary logical states. [Figure 9f] This diagram shows a schematic representation of the phase expansion due to the loss of consecutive bosons for the definitions of two exemplary logical states. [Modes for carrying out the invention]

[0016] The quantum states of bosons within a resonator can be modeled as a quantum harmonic oscillator system. Any possible state of a boson within a resonator can be represented by a distribution across specific excitation levels (known as the basis for the number of boson states or the basis for the Fock states). These states can also be represented by quasi-distributions in two-dimensional phase space, given by the Wigner transform of the state wavefunction.

[0017] It is known that the lowest energy state is the vacuum state. In phase space, the Wigner transform of the lowest energy state is represented by a circular quasi-Gaussian distribution at the origin of the two orthogonal phases of the Wigner transform.

[0018] Applying a displacement operator to a vacuum state yields a coherent state, which in phase space is identical to the vacuum state until it is deformed (translated).

[0019] This displacement can be parameterized by a complex number α, where the real part of α represents the displacement along the horizontal orthogonal phase, and the imaginary part represents the displacement along the vertical orthogonal phase.

[0020] All coherent states with the same absolute value of α lie on a circle of radius |α|, and all of them represent states with the same mean boson number (therefore, <n>=|α| 2 ).

[0021] A cat state can be defined as any superposition of any number of coherent states that lie on a circle centered at the origin of the phase space and therefore share the same value |α|.

[0022] Specifically, an n-cat state is a cat state formed by the superposition of n states uniformly distributed along the circumference of a circle with radius |α|.

[0023] Figure 1A shows an embodiment of a quantum subsystem that provides a cat state having two coherent states, which are represented as equidistant points around a circle in phase space. Such a cat state can simply be called a 2-cat.

[0024] Long-lived 2CAT states can be achieved using a stabilization mechanism with an external EM drive. Such encoding can be robust against so-called bit inversion errors (pure state |0> becomes |1> or vice versa). However, such encoding may not be robust against another significant error channel, namely phase inversion (a given specific superposition of states, e.g., |0>+|1>, is converted to another superposition, e.g., |0>-|1>), under the influence of excitation losses to the environment (e.g., single-photon loss). Therefore, while addressing bit inversion type errors, one of the two major error channels, can reduce the need for physical redundancy, the need for physical redundancy remains. Thus, it is still necessary to address both types of errors, bit inversion and phase inversion, simultaneously.

[0025] One possible cause of error in a boson cat state is a single-photon loss, where the state of the system after this undesirable interaction also represents a valid logical state. In such cases, the error may not be detectable from the logical state alone. That is, a typical qubit state is given by:

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[0026] One possible approach to circumvent this problem is to use a cat state with more coherent components.

[0027] Figure 1B presents an exemplary embodiment that provides a cat state in which the quantum subsystem has four coherent states instead of two. For ease of later reference, we will refer to the cat state with four coherent components as the "four-legged cat" or simply 4-cat, and the cat state with two coherent components as the "two-legged cat" or simply 2-cat. This approach faces two entirely different challenges. First, stabilizing these 4-cat states appears to be far more difficult than stabilizing the 2-cat state. Second, while the encoding of the 4-cat can theoretically distinguish a single error, this requires monitoring more frequently than the timeframe corresponding to the probability of a quasi-continuous boson loss. For example, in one embodiment, if a single error is missed, a scenario may arise where a second error causes a phase inversion error, and therefore becomes undetectable.

[0028] Here, it may be noticed that the second problem is somewhat conditional with respect to the fact that the four - cat logical encoding has rotational symmetry. Thus, in some embodiments, it may be preferable to select an N - cat logical encoding that does not have rotational symmetry, i.e., for which any photon losses, when n < N, are outside the logical basis and thus detectable and, in some cases, correctable states.

[0029] This may specifically be the case for a three - cat logical encoding, which seems to be particularly promising as presented below, and a schematic embodiment of which is presented in FIG. 1C. The first logical state can be associated with two of the coherent components, and the second logical state can be associated with the other coherent component (or a linear combination of coherent components).

[0030] It has been found that the rotation - asymmetric condition can be satisfied for other N - cat logical encodings where N is greater than 2, including the case of N = 4. The non - rotation - symmetric condition necessarily exists, for example, when N is also a prime number, but may also necessarily exist in other relatively simple ways, some of which will be detailed below. In other words, in the following specifications, we will consider an encoding scheme in a physical system where N>2 and a richer set of N distinct quantum states can be represented with the selected configuration not showing rotational symmetry. The logical qubit can be defined in this way. Further, in the following specifications, we will consider techniques that enable the execution of operations on qubits such that errors that may arise from undesirable interactions with the environment that cause photon losses can be corrected.

[0031] FIG. 1 presents a) a two - cat state, b) 4 a cat state, c) 3 a phase - space representation of a cat state, showing the superposition of coherent states (large circles) and the interference fringes generated by the coherent states.

[0032] One weakness common to 2CAT and at least some higher-order encoding schemes can perhaps most commonly be summarized as the CAT state, in which the logical state is encoded, being constructed such that when a boson loss occurs, the resulting change in state is indistinguishable from the initial logical subspace. In such constructions, simply checking the state may not detect the occurrence of a boson loss. One common technique to address this problem is to use a construction in which one (or ideally one or more) boson losses from the initial state result in a logical subspace that is clearly different from the initial logical subspace.

[0033] One exemplary method for implementing this general technique is to choose a coding scheme that does not exhibit rotational symmetry. In fact, in such a configuration, the loss of a boson from the initial state can result in a logical subspace that is clearly different from the initial logical subspace, making error detection possible. A logical subspace can be segmented into two logical states using a segmentation scheme. Several exemplary segmentation schemes that do not exhibit rotational symmetry are presented below, such as a segmentation scheme that uses prime number coherent components, or a segmentation scheme that encodes one logical state of one of the coherent components and the other logical state of a combination of the other coherent components. The simplest case is probably the CAT coding scheme with three coherent states, where the first logical state of the first coherent state is encoded, and the second logical state of a combination of the other two coherent states is encoded. While this simplest case may be preferred in some embodiments for several reasons, such as making stabilization easier, other segmentation schemes will be understood in which the loss of bosons results in distinct logical substates that are thus detectable. It will be noted that some alternative and preferable encoding schemes can be obtained, for example, from a careful selection of linear combinations chosen to define logical states that break the symmetry of the otherwise symmetric coherent components in a cat state.

[0034] Before considering further exemplary embodiments, we will begin by examining a first example based on a 3-CAT configuration.

[0035] Exemplary Example Attribution of a coherent state to a logical state Here, we will detail an exemplary embodiment based on a quantum subsystem for N=3 (which can also be called a quantum trit or cubic cat state) that is segmented by selecting a single coherent state as logic |0> and the other two combinations as logic |1> (or vice versa).

[0036] We begin by defining the states of the two logical qubits. When N=3, there are three originally coherent states, namely the following: |α>, |αe 2πi / 3 >, |αe -2πi / 3 > Here, α is a positive real number, usually called the "size" of the cat, because in the topological representation, it corresponds to the radius of the circle in which each coherent component state (often called a "foot") of the cat state is located. From these three coherent states, a valid segmentation is selected such that the first of these three states represents the logical state |0> and the other two represent the logical state |1>. It is obvious that logical |0> in this case is defined as follows:

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[0037] To define the logical state of |1>, we must make a choice. In this example, for clarity, the following choice is made.

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[0038] In further demonstration, we define a third quasi-orthogonal state that has proven useful. That is,

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[0039] Let's start with a typical logical qubit state.

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[0040] In this model, if the system suffers a 3-photon loss, the state returns exactly to the system's original state. This means that no matter how many photon losses occur, there are only three possible states to which the system can cycle. Most importantly, these other two states are not in the logical basis because they relate to a third quantum trit state |2>. Thus, the other two states are distinguishable and do not break the relative values ​​of u, v, and Φ, which carry the logical quantum information that needs to be preserved. This works thanks to the process of 1) finding a suitable segmentation of the original coherent state into two subsets that avoid rotational symmetry in the phase space representation, and 2) defining two logical states of these subsets that are not trivial, but quasi-orthogonal or preferably orthogonal.

[0041] Stabilization of the coherent state To reliably extend the lifetime of quantum information encoded in qubits, it may be preferable to implement a stabilization process. We propose a specific method for generating a Hamiltonian with desired properties to stabilize a three-legged cat. The proposed form of the Hamiltonian in a rotating frame is given below.

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[0042] The fundamental principle behind this proposal is that the Hamiltonian, along with the three-photon pump and detuning terms, allows for the use of the naturally occurring Kerr term.

[0043] To understand how this works, rewriting the Hamiltonian in the displaced frame as follows may be useful to see how, with the right choice of parameters, certain terms can be ignored, and as a result, vacuum becomes an eigenstate in this frame.

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[0044] Returning to the first section, it is described as follows: α = |α|e iθ

[0045] Next, it will proceed as follows:

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[0046] If this is true, the first term can be rewritten as follows:

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[0047] This is a quadratic function of |α|, and real roots are allowed if the following determinant holds.

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[0048] in particular, |ε3|<<|Δ| In that case, it will be as follows:

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[0049]

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[0050] Furthermore, when the condition regarding the absolute value of α (as described above) is taken into consideration,

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[0051] The proposed Hamiltonian stabilization is presented in metapotential representation in Figure 2. Note that the metapotential remains quasi-circular around its minimum value, which may help avoid compressive deformation in the stabilized state.

[0052] Operations on logical qubits The operations to be performed on the proposed cubic cat qubits depend to some extent on the choice of the complete processor architecture. There are several competing architectures. Exemplary examples of two of them, namely quantum annealers and gate-based quantum processors, will be presented below.

[0053] Depending on the architecture type and the type of quantum subsystem used as the basis for the logical states, the details of the embodiment of a quantum processor can vary considerably from one example to another. However, many architectures and types will generally require the use of two or more quantum subsystems to provide the logical states. In fact, as shown in Figure 3, a typical quantum processor would require at least two quantum subsystems interconnected to interact with each other, in interactions that typically involve entanglement, where the coherent states of the quantum subsystems are interconnected. The type of quantum subsystem will vary depending on the architecture. Most quantum systems, embodied as boson-based quantum processors, require some form of resonator to provide the coherent states, which will be driven by the quantum subsystems using some form of driving hardware that can control some of the bosons of the quantum subsystems. The driving hardware is controlled by a component referred to herein as a controller and is typically provided in the form of a classical computer. The quantum subsystems are usually cooled to very low temperatures and isolated from the environment. In quantum annealing type architectures, the quantum subsystems can be interconnected so that they can operate directly with each other. In gate-based quantum computing, quantum subsystems are typically interconnected via couplers, which are used to selectively control the interactions between quantum subsystems. The coupler is also a quantum subsystem, driven by driver hardware, which can be controlled by the same controller for convenience, and whose coherent states provide states that can interact through the coupler. Here, we will consider an exemplary method that can perform quantum computing using a 3CAT configuration.

[0054] A. Quantum annealing (QA) In QA, qubits are operated on continuously from the start to the end of the computation. The Hamiltonian applied to each physical qubit is continuously modified, starting from a simple initial Hamiltonian intended to be easily prepared, and then progressively modified to its final form representing the problem to be solved.

[0055] The three key steps of the process can be distinguished as preparation (or initialization), deployment, and reading.

[0056] Initialization: Initialization can be performed separately for each individual physical qubit, and in the simplest example, the parameters can be made identical for all qubits. In most cases, qubits are initialized to the |+> state (|+>=|0>+|1>). In the case of the proposed cubic cat coding, this can be achieved by setting the cavity to a vacuum state and gradually increasing the active parameters of the proposed Hamiltonian (i.e., the drive parameters). If the initial state needs to be different from the simplest case above, more general initialization methods can be used (for gate-based models, this will be discussed in the next section).

[0057] Expansion: The Hamiltonian is gradually modified during the expansion phase to represent the problem to be solved. Non-trivial problems involve the entanglement of any given qubit with at least one other qubit. In most cases, problems in the Hamiltonian involve only entanglement terms of the following form:

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[0058] Readout: The logical states of a qubit can be read out by projection or quantum non-destructive measurement, which distinguishes between coherent states with different phases through the interaction of a parametric beam splitter between the cubic transmon and the readout resonator.

[0059] B. Gate-based model (GB) In gate-based architectures, qubits only need to be computed in discrete time when a logical operation involving one or more qubits is performed. In practice, in many proposed embodiments, qubits are computed (or at least monitored) quasi-continuously for error correction purposes.

[0060] In one proposed embodiment, the need for any continuous operations or monitoring of the qubits may be obscured. Error correction operations only need to be performed in discrete time when a particular type of logical operation should be performed on one or more qubits. This method, of course, does not preclude continuous monitoring for other purposes if necessary.

[0061] Here again, using a method similar to that of the QA architecture, we distinguish between the three key steps of the computation process in the GB architecture: preparation (or initialization), cleaning (error removal) for computation, and reading.

[0062] Cleaning for computation: Since initialization and reading are both essentially substeps of this step, we will begin by describing the second step of the process.

[0063] The step referred to as "cleaning" in this specification may be necessary when performing operations that cannot be described by the multiplication of Z operators, since the Z operator may not be affected by photon loss.

[0064] For other operations, restoring the original state |φ> can only be achieved by using additional physical resources, i.e., auxiliary resources, the simplest of which is a single physical 2CAT qubit. The main idea is as follows: 1) Apply an entanglement operation to the quantum trit / auxiliary system so that these two elements become entangled and logical qubits are incorporated into both. 2) Reset the quantum trits so that the logical qubits are completely transferred to the auxiliary system. 3) Immediately apply reverse rotation to convert the logic qubit back into a quantum trit, protecting it from the harmful effects of photon loss. 4) Reset the auxiliary systems.

[0065] A key element of the entanglement operation in step 1) above is realized as a modified controlled phase gate, with the auxiliary system as the target and the cubic cat as the control unit. A description of the series of exemplary operation steps involved in this process is presented in the following section.

[0066] Initialization: Initialization is performed by first preparing the desired logical state with the auxiliary qubits, and then performing steps 3) and 4) similar to those above (cleaning for operation).

[0067] Readout: Readout is performed by applying steps 1) and 2) above (cleaning for operation), and then performing a readout on the auxiliary qubit.

[0068] Advanced explanation of the steps for performing cleaning for calculations

[0069] The cleaning operation can be adapted to various specific embodiments based on the general principles presented above. Here, we will consider possible adaptations of the three CATs presented above to specific embodiments and present one exemplary embodiment in detail.

[0070] From the above considerations, since a three-photon loss returns the system to its original state, the objective can be limited to recovering from either a one-photon loss or a two-photon loss. Therefore, the main error channel cycles between three distinct states that store the embedded qubit information, thereby eliminating the need to continuously monitor photon loss and only recovering when necessary, i.e., by using a non-conservative gate.

[0071] The cleaning operation can be adapted to various specific embodiments based on the general principles presented above. Here, we will consider possible adaptations of the three CATs presented above to specific embodiments and present one exemplary embodiment in detail.

[0072] In Figure 4, to represent the tensor product of the auxiliary qubit and the quantum trit, the cartoon Wigner diagram of the two legs of the auxiliary 2-cat qubit is shown as two large outlined red circles, each containing the cartoon Wigner diagram of the corresponding quantum trit.

[0073] Immediately before the correction, the quantum trit is in state Ψ, one of only three possible states. That is,

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[0074] Also, as mentioned above,

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[0075] One approach would be to simply copy the state of the quantum trit, measure one of the three possible states, and apply an appropriate correction to the original quantum trit. Unfortunately, this is prohibited by the no-cloning theorem for quantum states. However, moving quantum information from one subsystem to another is not prohibited. Also, while a logical qubit would have little meaning simply moving around a complete quantum trit, it can move from inside a physical quantum trit onto an additional physical qubit (referred to as an ancilla or ancillary qubit) coupled to the physical quantum trit, leaving only unwanted phase relaxation, and then reset the original quantum trit to eliminate unwanted phase relaxation caused by noise (see Fig. 4).

[0076] The first operation step is to prepare an ancillary qubit in the |+> state by a rotation about the x-axis by πi / 2 while the ancillary qubit is not yet coupled to the quantum trit (lines 2 to 3 of Fig. 4). After this step, the combined qubit and trit system is in the following state. |+> b |Ψ> t Here, the subscripts b and t refer to the qubit and trit, respectively.

[0077] The next step, which is also the main step of this process and is described analytically as follows, is to entangle the quantum trit with the ancillary qubit by a particular qubit-trit controlled-phase gate CZ = ZZ(π / 2). [Number]

[0078] The combined state is converted to the following:

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[0079] This final step is shown in the third and fourth lines of Figure 4.

[0080] In the specific case where the auxiliary qubit is in a two-legged cat state, this operation can be realized by performing the following auxiliary system / quantum trit interaction Hamiltonian.

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[0081] The final step consists of resetting the quantum trit to its |0> state, which is facilitated by a prior -π / 2 x rotation of the auxiliary qubit (lines 4 to 5 and 5 to 6 in Figure 4).

[0082] Modes and Interactions Figure 5 shows a schematic diagram representing modes and interactions in an exemplary embodiment in which a 2-CAT quantum subsystem is added to each 3-CAT subsystem for the cleaning described above. In the photon-type embodiment, the mode providing the 3-legged CAT can interact with an auxiliary mode providing a 2-legged CAT state (2-CAT) through a time-controlled beam splitter interaction. The 2-legged CAT state can be read out through a time-controlled beam splitter interaction with a readout mode.

[0083] Example circuit Figure 6 shows an exemplary circuit for carrying out the process described above. The three-legged and two-legged cat states reside in two cubic transmons, each consisting of a small Josephson junction in parallel with a larger junction of N≧2, and a large shunt capacitance. A finite magnetic flux is supplied to the junction loop to realize second-order nonlinearity. The magnetic flux can be supplied by individual flux lines (as shown in the figure) or by a single flux line. Individual drive ports can excite the cubic transmons with one-photon and three-photon pumps (one-photon and two-photon pumps) and stabilize the three cats (two cats). The two cubic transmons are capacitively coupled, resulting in a fixed beam splitter interaction. If both modes are greatly detuned from each other, the beam splitter interaction can result in a static, undesirable, dispersive ZZ-type interaction. This interaction can be suppressed by adding a continuous drive to either of the cubic transmons. The desired time-controlled beam-splitter interaction between both cubic transmons is parametrically activated by driving either mode by the frequency difference of the cubic transmons. Quantum non-destructive readout of a two-legged cat state is performed by parametrically activating the beam-splitter interaction between the corresponding cubic transmon and the readout resonator by driving the cubic transmon by the frequency difference between the cubic transmon and the resonator.

[0084] In one example, the circuit can be equipped with the following drives: Stabilizing 3CAT: One-photon drive at cubic transmon frequencies Stabilization of 3CAT: 3-photon drive at three times the cubic transmon frequency. Stabilization of 2CAT: Two-photon drive at twice the cubic transmon frequency Calculations on 2CAT: One-photon drive at cubic transmon frequencies 2CAT readout: Parameterized drive based on the difference between the 2CAT cubic transmons and the corresponding resonator. Interaction between 3CAT and 2CAT: Parameterized drive at the difference in cubic transmon frequencies for time-controlled ZZ-type interaction. Interaction between 3-cat and 2-cat: Parameterized drive to suppress static ZZ-type interaction

[0085] Exemplary Alternative Examples Bosons other than photons The proposed scheme can be implemented in alternative architectures for modes where the elementary excitations are other types of bosons. The most noteworthy candidate is a phonon-based architecture. One clear advantage of such an architecture is that phononic resonators with lifetimes reaching 1 second have been demonstrated. Another possible embodiment would use magnetostatic modes where the magnons are bosonic excitations. One major challenge in magnon-based architectures is their short lifetime, approximately several hundred nanoseconds. In both alternative physical embodiments, the nonlinearity required in the proposed scheme is likely to be achieved by coupling the bosonic modes to a nonlinear superconducting circuit.

[0086] Potential for stabilizing cats with N > 3 legs The second-order linearity (Pockels) and third-order nonlinearity (Kerr) present in cubic transmons provide, in some cases, the necessary conditions for designing N-photon drives where N > 3. However, it will be noticed that the efficiency of such drives may decrease as N increases.

[0087] Figures 7a to 7j are a set of schematic diagrams representing coding examples that can form a basis for qubits. In a general form, it has been proposed to perform coding on N-legged cat (or N-cat) states for N>2. These N individual states are all equidistant on the circumference when represented in the phase space of the resonator cavity.

[0088] Each of the N individual states is segmented into two subsets, each represented by either a white or black circle, and each subset represents one of the two logical states of a qubit. The logical states are constructed to be rotationally asymmetric; that is, the angle by which a logical state can be rotated back to the same configuration is never less than one rotation. Figures 7a to 7f present a method for realizing this rotational asymmetry based on the assignment of a subset of states to a given state, which we will call asymmetry by subset configuration. Figure 7j presents an example where the rotational asymmetry is due to the assignment of zero coefficients to a subset configuration that would otherwise be rotationally symmetric, and can be called asymmetry by coefficient configuration. First, we will begin by discussing how to realize rotational asymmetry based on subset configuration.

[0089] For example, if N is a prime number greater than 2, this condition is automatically taken into account in all configurations where segmentation between two subsets is possible. However, note that N being odd does not guarantee the absence of rotational symmetry, as shown in Figure 7i. Figures 7g and 7h also represent segmented configurations in which rotational symmetry exists. Figures 7a through 7f present various examples in which rotational symmetry does not exist.

[0090] Various methods or rules can be implemented to facilitate or automate the realization of segmentation without rotational symmetry. For example, regardless of whether N is a prime number or not, a segmentation configuration without rotational symmetry can be realized by selecting two consecutive subsets, as shown in Figures 7a and 7c. A simpler segmentation can always be obtained by selecting just one state to represent one of two logical states and the remaining N-1 states to represent the other logical state, as in the configurations presented in Figures 7c and 7f. However, selecting this configuration for relatively large N may make it more difficult to achieve the additional requirement of approximate orthogonality of both logical states, which will be discussed below. When N is odd, another suitable segmentation configuration can be obtained simply by alternating the segmentation, as shown in Figure 7d.

[0091] In general, especially when stabilization may be an issue, it may be preferable to choose a small value of N for simplification. For example, it may be preferable to choose an N value of 3, 4, or 5. N=3, as shown in Figure 7f, may be more preferable in some embodiments and is selected based on achieving the detailed exemplary embodiments presented below.

[0092] Once an appropriate subset is selected, each quantum state (|0> and |1>) can be defined by any non-trivial linear combination of states that constitute each subset of states, where complex coefficients are allowed and the selected states can be appropriately normalized. In doing so, it is necessary to ensure at least approximate orthogonality, if not exact orthogonality, between the two logical states.

[0093] While it is possible to adapt the coding of logical states in a rotationally symmetric segmented configuration, the symmetry is broken by a specific selection of coefficients in the linear combination. A simple example of such a transformation is shown in Figure 7j, where the coefficient 0 is assigned to the state marked with the symbol "X", and this 6-cat rotationally symmetric segmented configuration is effectively transformed into a 3-cat asymmetric configuration. Both concepts can be combined, and the asymmetric configuration can further be assigned one or more zero coefficients and remain rotationally symmetric.

[0094] It will be noticed that other embodiments are also possible. This concept can perhaps be described more generally by referring to Figures 8a-8e, which schematically represent the phase shift from the initial state. In Figure 8a, two distinct logical states are defined in the phase space representation of the figure as i) a superposition of groups of white dots and ii) a superposition of groups of black dots. The definition of each logical state involves assigning complex numbers, i.e., amplitude and phase, to each coherent state (these can also be zero, in which case each coherent state is excluded from the definition of a logical state). As is known, the sum of the squares of the complex amplitudes of each logical state is equal to 1, and the inner product of both logical states is minimized. The initial state is a given superposition of logical states that represents logical information.

[0095] In Figure 8b, the phase of each coherent state in the initial state is represented by the hands of a clock. Phase shifts can occur in some types of undesirable or unexpected interactions with the environment, i.e., interactions that do not conform to the interactions driven by control commands. Phase shifts are influenced in various ways by different individual coherent states. For example, Figure 8c shows a scenario in which the first coherent state (on the right) remains unaffected by a phase shift, while the phase change, represented by the changing position of the hands of the clock, increases proportionally from the first coherent state to the next coherent state in a counterclockwise order around the circle.

[0096] As demonstrated in this example, if various coherent states associated with each logical state remain in phase with each other, but are out of phase with coherent states associated with other logical states, logical information is lost. This is undesirable, but it can be avoided by defining the logical states in a way that prevents this from happening. Many different techniques can be used to ensure this does not occur, and many of these techniques are presented above, forming the basis for numerous examples.

[0097] However, in other embodiments, other techniques can be used. For example, consider a scenario in which S is the set of all n coherent states of a system, A and B are exact 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), the first logical state consists of elements of A (which may include all elements of A, or any subset of A), and the second logical state consists of elements of B (which may include all elements of B, or any subset of B). In such a scenario, an uncorrectable error may occur if A and B are selected as follows: i) Both A and B are rotationally symmetric and equally spaced subsamples of set S (e.g., states 1 and 0 in the upper row, state 0 in the lower row), and ii) B can be obtained by rotating A (B is simply A rotated, and vice versa). Therefore, in order to avoid potential problems, A and B can be selected as follows: i) At least one of the logical states is not composed of a subset of a rotationally symmetric subset of S, or ii) If both logical states are composed of a subset of a rotationally symmetric subset of S, then the logical states must not be composed of subsets of two rotationally symmetric subsets of S that are identical depending on the rotation.

[0098] In contrast to the scenarios presented in Figures 8a to 8e, another scenario using the same logical representation is presented in Figures 9a to 9f. In the scenarios of Figures 9a to 9f, the logical states are defined to either become one or more modified states (e.g., Figures 9c, 9d, 9e, 9f) outside the logical subspace, while retaining logical information, by limiting any of the possible phase shift increments from the initial state in Figure 9b, as presented in Figures 9c, 9d, 9e, and 9f, or returning to the initial state. In fact, in each of the intermediate phase shifts represented in Figures 9c to 9f, the coherent states of each logical state are out of phase with the coherent states associated with other logical states, but not in phase with each other.

[0099] The definition of a logical state can include assigning a distinct combination of complex numbers to each coherent state. In the scenarios presented in Figures 7a-7i, 8a, and 9a, the complex numbers are weighted to 1 or 0, and a combination of coherent states weighted to 1 in the first logical state definition is weighted to 0 in the second logical state, and vice versa. In the scenario presented in Figure 7j, some coherent states are not used because they are weighted to zero in both logical state definitions. In various embodiments, the exact selection of the combination of complex numbers associated with each coherent state and defining the two logical states may vary, and logical states may share some or all of the coherent states, and are weighted in various ways, including weighting to values ​​other than 1, simply to minimize the inner product of the logical states. Similarly, in Figure 9b, all coherent states are synchronized in the initial state, but it will be understood that coherent states may be asynchronous to each other in the initial state.

[0100] It should be understood that the terms “computer,” “classical computer,” or “controller” as used herein should not be interpreted restrictively. “Computer” is used rather broadly, generally referring to a combination of one or more processing units of any form and some form of memory system accessible by the processing units. “Controller” is used broadly, generally referring to a device that performs control functions, which may be a computer or another type of device. A memory system is used when the computer may be of a non-temporary type. The singular use of the term “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 computers are on-premises, remote, or distributed. Furthermore, the term “computer” as used herein includes, within its scope, the use of a partial function of a given processing unit.

[0101] Processing units can be implemented in several forms, including, to give a few examples, general-purpose microprocessors or microcontrollers, digital signal processing (DSP) processors, integrated circuits, field programmable gate arrays (FPGAs), reconfigurable processors, and programmable read-only memory (PROMs).

[0102] The memory system may include a suitable combination of any suitable type of computer-readable memory, located either internally or externally, and accessible to the processor directly or wirelessly via a network such as the Internet, either by wire or wirelessly. Some examples of computer-readable memory include random-access memory (RAM), read-only memory (ROM), compact disc read-only memory (CDROM), electro-optical memory, magneto-optical memory, erasable programmable read-only memory (EPROM), electrically-erasable programmable read-only memory (EEPROM), and ferroelectric RAM (FRAM®).

[0103] A computer may have one or more input / output (I / O) interfaces that enable communication with a human user and / or with another computer via associated input, output, or input / output devices such as a keyboard, mouse, touchscreen, antenna, or port. Each I / O interface can enable the computer to communicate with other components and / or exchange data, access and connect to network resources, provide services to applications, and / or run other computer processing applications by connecting to one (or more) networks capable of transmitting data, including, to name a few, the Internet, Ethernet®, plain old telephone service (POTS) lines, public switched telephone networks (PSTN), integrated services digital network (ISDN), digital subscriber lines (DSL), coaxial cables, optical fibers, satellites, mobile phones, wireless (e.g., Wi-Fi, Bluetooth, WiMAX), SS7 signaling networks, fixed lines, private network areas, and wide area networks.

[0104] It will be understood that a computer can perform a function or process through hardware, or a combination of both hardware and software. For example, hardware may include logic gates, which are provided as part of a processor's silicon chip. Software (e.g., applications, processes) may take the form of data, such as computer-readable instructions, stored in non-temporary computer-readable memory accessible to one or more processing units. The expression "configured to" in reference to a computer or processing unit relates to the existence of hardware, or a combination of hardware and software, that is capable of operating to perform the relevant function.

[0105] For clarity, the examples described and illustrated above are for illustrative purposes only. The scope is indicated by the attached claims.< / n>

Claims

1. A quantum computing system, A quantum subsystem that can be driven to provide a boson cat state having N coherent states, where N is 3, and the N coherent states define a computation space. Drive hardware that is operably connected to drive the quantum subsystem to provide the cat state of the bosons and to control the number of bosons in the quantum subsystem, A controller operably connected to control the drive hardware according to control commands, Equipped with, The control instruction is defined as a given superposition of two distinct logical states representing logical information, and includes a function by which the controller generates an initial state of the boson's cat state, which extends across a logical subspace within the computation space. When the initial state of the cat state of the boson is generated, a phase shift from the initial state resulting from the loss of any number of bosons in the quantum subsystem occurring outside of controlling the number of bosons, i) To bring about a modified state outside the logical subspace while retaining the logical information, or ii) Return to the initial state described above. A quantum computing system that is constrained in such a way.

2. The quantum computing 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, and when the cat state of the boson rotates around the circle in the phase space due to the loss of any number of bosons, the logic state becomes rotationally asymmetric in the phase space.

3. The quantum computing system according to claim 2, wherein rotational symmetry is broken by giving a coefficient of zero to at least one of the coherent states.

4. The aforementioned quantum subsystem is a coherent state |α>, |αe ２πｉ／３ >, |αe －２πｉ／３ The quantum computing system according to claim 3, which is driveable to provide, wherein a first state among the logical states is encoded as a first coherent state among the coherent states, and a second state among the logical states is encoded together with two other coherent states.

5. The aforementioned boson is a photon, The quantum subsystem is a superconducting circuit comprising a resonator having a resonator frequency, The drive hardware comprises a three-photon drive having a drive frequency, The aforementioned boson's cat state is Hamiltonian [Math 1] Driven by the drive hardware based on, where K is the amplitude of the Kerr nonlinearity, and ε ３ The quantum computing system according to claim 4, wherein Δ is the amplitude of the three-photon drive, and Δ is the detuning between the drive frequency and the resonator frequency.

6. The aforementioned controller [Math 2] The quantum computing system according to claim 5, configured to control the drive hardware in such a manner.

7. The quantum computing system according to claim 6, further comprising an auxiliary subsystem that can be driven to provide a cat state of a boson having two coherent states, wherein the auxiliary subsystem is selectively connectable such that the coherent states of the auxiliary subsystem are selectively entangled or disentangled from the coherent states of the quantum subsystem.

8. The quantum subsystem comprises a first cubic transmon having a first cubic transmon frequency, The auxiliary subsystem comprises a second cubic transmon coupled to the first cubic transmon and having a second cubic transmon frequency, and a readout resonator coupled to the second cubic transmon and having a resonant frequency. The drive hardware includes a one-photon drive connected to the first cubic transmon and operating at the first cubic transmon frequency, a three-photon drive connected to the first cubic transmon and operating at three times the cubic transmon frequency, a two-photon drive connected to the second cubic transmon and operating at twice the cubic transmon frequency, a one-photon drive connected to the second cubic transmon and operating at the second cubic transmon frequency, and a connection between the first cubic transmon and the second cubic transmon. The quantum computing system according to claim 7, comprising: a first parametric drive operating at the difference between the first cubic transmon frequency and the second cubic transmon frequency for a continuous, time-controlled zz-type interaction; a second parametric drive connected between the first cubic transmon and the second cubic transmon and configured to suppress the static zz-type interaction; and a third parametric drive coupled to the second cubic transmon and driven at the difference between the second cubic transmon frequency and the readout resonator.

9. A method for providing a cat state of a boson having N coherent states in a quantum subsystem, wherein N is 3, and the N coherent states define a computation space, and the method is The steps include generating an initial state of the boson's cat state in the quantum subsystem via drive hardware that is operably connected to drive the quantum subsystem to host the cat state of the boson and controls the number of the bosons in the quantum subsystem, The initial state is a given superposition of two distinct logical states representing logical information, the two distinct logical states are spread across a logical subspace within the computation space, and the two logical states are phase-shifted from the initial state resulting from the loss of any number of bosons in the quantum subsystem occurring outside of controlling the number of bosons. i) While retaining the logical information, bring about one or more modified states outside the logical subspace, or ii) A method defined such that it is constrained to return to the initial state.

10. The method according to claim 9, wherein the N coherent states can be represented as points around a circle in the phase space of the quantum subsystem, and when the cat state of the boson rotates around the circle, the logic state becomes rotationally asymmetric due to the phase shift.

11. The method according to claim 10, further comprising the step of detecting a logical error in the coherent state, which includes the step of detecting the corrected state.

12. The method according to claim 11, further comprising the steps of: driving a cat state of a boson having two coherent states in an auxiliary subsystem; moving the logic state from the cat state of the boson of the quantum subsystem to the cat state of the boson of the auxiliary subsystem; resetting the coherent state of the quantum subsystem; and returning the logic state from the cat state of the boson of the auxiliary subsystem to the cat state of the boson of the quantum subsystem.

13. The aforementioned boson is a photon, The cat state of the boson in the quantum subsystem is provided in a first photon mode, and the cat state of the boson in the auxiliary subsystem is provided in a second photon mode. The method described above is The steps include: the interaction between the coherent state of the quantum subsystem and the coherent state of the auxiliary subsystem is performed by the interaction of a time-controlled beam splitter; The method according to claim 12, further comprising the step of reading the cat state of the boson of the auxiliary subsystem by interaction of a second time-controlled beam splitter.

14. The aforementioned boson is a photon, The quantum subsystem is a superconducting circuit comprising a resonator having a resonator frequency, The drive hardware comprises a three-photon drive having a drive frequency, The step of generating the initial state includes the step of setting the cavity of the quantum subsystem to a vacuum state and the Hamiltonian [Math 3] This includes a step of gradually increasing the drive parameter, Here, K is the amplitude of the Kerr nonlinearity, and ε ３ The method according to claim 10, wherein Δ is the amplitude of the three-photon drive, and Δ is the detuning between the drive frequency and the resonator frequency.

15. The method according to claim 10, further comprising the steps of: 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 so that the logical state is incorporated into both; resetting the quantum subsystem so that the logical state is provided only from the auxiliary subsystem; using reverse rotation to return the logical state from the auxiliary subsystem to the quantum subsystem; and resetting the auxiliary subsystem.

16. The method according to claim 10, further comprising: generating a cat state of a boson having two coherent states in an auxiliary subsystem; applying an entanglement operation between the quantum subsystem and the auxiliary subsystem so that the logical state is incorporated into both; resetting the quantum subsystem so that the logical state is provided only from the auxiliary subsystem; using reverse rotation to return the logical state from the auxiliary subsystem to the quantum subsystem; and resetting the auxiliary subsystem.

17. The method according to claim 10, further comprising the steps of: driving a cat state of a boson having two coherent states in an auxiliary subsystem; applying an entanglement operation between the quantum subsystem and the auxiliary subsystem so that the logical state is incorporated into both; resetting the quantum subsystem so that the logical state is provided only from the auxiliary subsystem; and performing a readout on the auxiliary subsystem.

18. The method according to claim 17, wherein the step of applying the entanglement operation is realized as a modified controlled phase gate in which the auxiliary subsystem is set as a target and the quantum subsystem is set as a control unit.

19. A control instruction stored in non-temporary computer-readable memory, wherein the control instruction is A function, when executed by a computer, that generates an initial state of a cat state of bosons having N coherent states in a quantum subsystem, where N is 3, the N coherent states define a computation space, the initial state is defined as a given superposition of two distinct logical states representing logical information, the two distinct logical states extend into a logical subspace within the computation space, the two logical states are defined such that the phase shift from the initial state resulting from the loss of any number of bosons in the quantum subsystem occurring outside of controlling the number of bosons, and at least one type of interaction between the cat state of bosons and the environment, not arising from the control instructions, results in the phase shift of the N coherent states. Definitions of the two distinct logical states, wherein the definitions constrain the expansion of the cat state of the boson from the initial state to one or more modified states outside the logical subspace while retaining the logical information, or back to the initial state, depending on any of the phase shifts. Control instructions, including those mentioned above.

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