Scalable optical quantum computing with hybrid resource states

Abstract

A system for scalable, fault-tolerant optical quantum computing includes multiple optical circuits, multiple photon-number-resolving detectors (PNRs), a multiplexer, and an integrated circuit (IC). In operation, the optical circuits generate output states by Gaussian boson sampling (GBS), and the PNRs generate qubit clusters based on the output states. The multiplexer multiplexes the qubit clusters and replaces empty modes with squeezed vacuum states to generate multiple hybrid resource states. The IC stitches the hybrid resource states into a high-dimensional cluster state containing states for fault-tolerant quantum computing.

Description

[Technical Field]

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to and benefit of U.S. Provisional Patent Application No. 63 / 084,994, entitled "Scalable Optical Quantum Computing with Hybrid Resource States," filed September 29, 2020, the entire contents of which are incorporated herein by reference.

[0002] Field FIELD OF THE DISCLOSURE

[0002] The present disclosure relates generally to the field of optical quantum computing, and more particularly to the generation of three-dimensional resource states such as boson qubits and squeezed states. [Background technology]

[0003] background

[0003] The ability to store and coherently manipulate quantum optical pulses is desirable for the development of long-distance quantum communication and quantum computing. Integrating these capabilities onto photonic chips, along with other quantum technologies such as entangled photon sources, is an important practical step toward implementing such applications. Summary of the Invention [Means for solving the problem]

[0004] overview

[0004] One or more embodiments described herein relate to generating three-dimensional resource states including boson qubits and squeezed states. In some embodiments, a system for scalable, fault-tolerant optical quantum computing includes a plurality of optical circuits, a plurality of photon-number-resolving detectors (PNRs), a multiplexer, and an integrated circuit (IC). In operation, the optical circuits generate output states by Gaussian boson sampling (GBS), and the PNRs generate qubit clusters based on the output states. The multiplexer multiplexes the qubit clusters and replaces empty modes with squeezed vacuum states to generate multiple hybrid resource states. The IC transforms the hybrid resource states (e.g., by stitching the hybrid resource states) into higher-dimensional cluster states that include states suitable for fault-tolerant quantum computing.

[0005] BRIEF DESCRIPTION OF THE DRAWINGS

[0005] The drawings are primarily for illustrative purposes and are not intended to limit the scope of the subject matter described herein. The drawings are not necessarily drawn to scale. In some instances, various aspects of the disclosed subject matter disclosed herein may be shown exaggerated or enlarged in the drawings to facilitate an understanding of various features. In the drawings, like reference numerals generally refer to like features (e.g., functionally similar and / or structurally similar elements). [Brief explanation of the drawings]

[0006] [Figure 1] FIG. 1 is a system diagram illustrating a system for generating a hybrid cluster state, according to one embodiment. [Figure 2A]

[0007] FIG. 1 illustrates an integrated photonic device implementing Gaussian Boson Sampling (“GBS”)-based preparation of non-Gaussian states. [Figure 2B]

[0008] FIG. 1 shows a simplified representation of a single GBS device. [Figure 2C]

[0009] FIG. 1 illustrates a state preparation device including GBS devices multiplexed in the spatial and / or time domain, according to one embodiment. [Figure 3A]

[0010] FIG. 1 illustrates the generation of a 1D qubit cluster in the time domain, according to one embodiment. [Figure 3B]

[0010] FIG. 1 illustrates the generation of a 1D qubit cluster in the time domain, according to one embodiment. [Figure 3C]

[0011] FIG. 3B illustrates a simplified representation of the 1D qubit cluster generator of FIG. 3A. [Figure 4]

[0012] 3B is a time-domain equivalent circuit for the 1D qubit cluster generator of FIG. 3A showing that the output is a 1D cluster state. [Figure 5A]

[0013] FIG. 1 illustrates the generation of a (1+1)D cluster state using multiple 1D cluster generators according to one embodiment. [Figure 5B]

[0014] 1 is an exemplary 2D chip layout for a (2+1)D cluster generator, according to one embodiment. [Figure 5C]

[0015] FIG. 5C illustrates a 3D cubic lattice produced using a 2D chip having the layout of FIG. 5B, according to one embodiment. [Figure 6A]

[0016] FIG. 1 shows an exemplary representation of two layers of a Raussendorf lattice separately. [Figure 6B]

[0017] 6B is a composite representation of two layers of the Raussendorf lattice of FIG. 6A. [Figure 6C]

[0018] FIG. 6B shows an exemplary chip layout including multiple controlled-Z (“CZ”) gates for generating the Raussendorf lattice of FIG. 6A. [Figure 7]

[0019] FIG. 10 illustrates node types for a state generator implementation using beam splitters instead of CZ gates, according to one embodiment. [Figure 8]

[0020] FIG. 8 is a schematic diagram of a passive system configuration chip layout for generating states, including the nodes shown in FIG. 7, according to one embodiment. [Figure 9]

[0021] FIG. 1 illustrates an exemplary graph transformation for passive composition of a general graph state, according to one embodiment. [Figure 10A]

[0022] FIG. 9 illustrates an equivalent circuit for the state shown in FIG. 8, according to one embodiment. [Figure 10B] FIG. 9 illustrates an equivalent circuit for the situation shown in FIG. 8, according to one embodiment. [Figure 11A]

[0023] 9A-9C illustrate equivalent circuits for the various types of entangled states shown in FIG. 8, according to one embodiment. [Figure 11B] FIG. 9 illustrates equivalent circuits for the various types of entangled states shown in FIG. 8, according to one embodiment. [Figure 11C] FIG. 9 illustrates equivalent circuits for the various types of entangled states shown in FIG. 8, according to one embodiment. [Figure 12]

[0024] FIG. 1 illustrates a principal unit cell of a 3D hybrid paired macro-node cluster state and steps for generating it, according to some embodiments. [Figure 13]

[0025] FIG. 10 illustrates an identity confirming the equivalence between a two-mode entangled state and a two-mode cluster state, according to one embodiment. [Figure 14]

[0026] FIG. 14 shows the equivalence of two-mode cluster states generated with and without a CZ gate, obtained from the identities of FIG. 13. [Figure 15]

[0027] FIG. 2 illustrates a circuit representation of a beam splitter network associated with a single macro node and its relationship to adjacent macro nodes, along with their equivalent circuit representations, according to one embodiment. [Figure 16]

[0028] FIG. 10 illustrates hybrid macro node 3D cluster state mode placement and edge disambiguation according to one embodiment. [Figure 17]

[0029] FIG. 1 illustrates the modularity of the system architecture of a GBS generation chip, according to some embodiments. [Figure 18]

[0030] 1 is a flow diagram illustrating a first method for generating a hybrid cluster state, according to some embodiments. [Figure 19]

[0031] 10 is a flow diagram illustrating a second method for generating a hybrid cluster state, according to some embodiments. DETAILED DESCRIPTION OF THE INVENTION

[0007] Detailed Description

[0032] While fault-tolerant optical quantum computing is an active area of ​​research, known systems have proven impractical to implement due to cost, size, and / or complexity. For example, some known quantum computing systems perform near-deterministic generation of non-Gaussian states, but do so at prohibitive experimental costs and require an impractical number of devices to achieve a desired level of reliability. Other known quantum computing systems perform direct generation of encoded qubit clusters, but are difficult to multiplex and expensive to implement.

[0008]

[0033] The embodiments described herein include system configurations for optical quantum computing that overcome the challenges of known systems described above by generating and manipulating three-dimensional resource states that include both bosonic qubits and squeezed vacuum states. Some embodiments leverage known methods for nondeterministic generation of bosonic qubits while simultaneously providing the benefits of continuous-variable quantum computing, for example, by implementing Clifford gates using squeezed states. Some system configurations described herein are implemented using two-dimensional circuits of integrated photonic chips that generate qubit cluster states in one time dimension and two spatial dimensions. Two-dimensional circuits facilitate a modular approach to quantum computing, as different photonic chips can be optimized for various aspects of a desired computing protocol. In some such implementations, a primary computational chip can be used for operation at ambient conditions, enabling scalable fabrication and operation of quantum computers.

[0009] introduction

[0034] On the way to building scalable, fault-tolerant quantum computers, photonics promises several advantages over competing platforms. These advantages may include: (i) the possibility of room-temperature computation, which allows scaling up to large numbers of qubits by employing (with minimal modifications) known silicon electronics and photonics technologies; (ii) inherent compatibility with communications technologies, which allows high-fidelity connections between multiple modules (i.e., photonic or other multiple quantum computing circuits) without the noisy conversion steps of other platforms; and (iii) inherent flexibility in the choice of error-correcting codes, including high-dimensional codes that use temporal degrees of freedom on the way to fault tolerance. These advantages motivate serious consideration of optical quantum computing system construction.

[0010]

[0035] Known optical quantum computing system configurations can be viewed as including two main classes. The first class of system configurations is based on the use of continuous variable (CV) cluster states with encoded qubits, taking advantage of the relative ease (compared to known techniques) of generating CV cluster states from squeezed light. In such system configurations, quantum information is encoded in bosonic qubits, such as the type of bosonic qubit proposed by Gottesman, Kitaev, and Preskill (hereinafter "GKP" qubits), and Clifford and non-Clifford operations are implemented using CV cluster and non-Gaussian resources, respectively. Further details regarding GKP encoding can be found in "Encoding a Qubit in an Oscillator," D. Gottesman, et al., Physical Review Letters A (64), 012310 (2001). This class includes the design and generation of CV cluster states in various lattices, such as the two-layer square lattice (BSL), double BSL, and modified BSL. Known implementations of system configurations within the first class often rely on near-deterministic generation of non-Gaussian states to encode information, act as non-Clifford gates, and correct CV errors. This reliance on near-deterministic generation of non-Gaussian states can lead to prohibitively high experimental costs. Multiplexing non-deterministic state generation procedures to generate states with near-unity probability 1-p implies that the number of state-generating devices scales as 1 / p, which becomes very large as p → 0.

[0011]

[0036] The second class of system configurations is based on directly generating clusters of encoded qubits. This class includes schemes developed for dual-rail encoding, cat-basis encoding, and GKP encoding. These schemes differ from the first class of schemes in that they are well suited to nondeterministic state generation of encoded qubits and the nondeterministic gates used to construct the cluster states. However, a challenge with these schemes is that each gate is ultimately implemented by consuming encoded qubits, which, due to their nondeterministic nature, have complex multiplexing requirements and therefore consume far more experimental resources than nodes for deterministically generated CV cluster states.

[0012]

[0037] In view of the above, it would be desirable to devise a hybrid scheme that leverages the advantages of both classes of optical quantum computing system configurations, for example, by implementing Clifford operations using CV resources while still remaining compatible with non-deterministic generation of encoded qubits. One or more embodiments described herein present just such a scheme.

[0013]

[0038] In addition to the advantages described above, the disclosed optical quantum computing system configurations exhibit several desirable features for scalability. One such desirable feature is that the system configurations may be suitable for full on-chip implementation, in contrast to known systems for CV optical quantum computing, which are typically optimized for free-space implementation. Full on-chip implementation is facilitated by the planar nature of one or more of the disclosed optical quantum computing system configurations, e.g., each qubit is connected to a small, fixed number of neighboring qubits, and there are a fixed number of intersecting channels per qubit. Another desirable feature is that the system configurations are modular, and the size and number of integrated photonic chips are independent of the desired circuit depth. Furthermore, one or more of the disclosed individual modules can be specialized to ensure compatibility with other technologies, unlike known optical quantum computing system configurations that include seemingly incompatible technology combinations (e.g., system configurations for dual-rail qubits). As an example, consider the challenge of achieving low-loss, high-speed reconfigurable optical switching at cryogenic temperatures. According to one or more embodiments of the optical quantum computing system configuration, the state generation module may be low-loss but not reconfigurable; the multiplexing module may exhibit less stringent loss constraints and thus be reconfigurable; and the computation module may support reconfigurable switches with relatively high loss (e.g., 0.1 dB to 10 dB loss) at room temperature. Furthermore, the computation module may enable operation at ambient temperature and pressure, thus enabling manufacturing scalability (e.g., via a complementary metal-oxide semiconductor (CMOS) process).

[0014]

[0039] From a theoretical perspective, one or more embodiments of the present disclosure encompass at least two novel features. First, we describe a planar system configuration for measurement-based quantum computing in the Raussendorf model using CV-encoded qubits (see the "Modular System Configuration" section below). Second, we describe a method for fault-tolerant quantum computing using hybrid resource states, in which some sites (i.e., optical pulses at a given location on a quantum system chip at a given time) are boson qubits and other sites are squeezed vacuum. Such resources can be generated using relatively few experimental resources. In the "Quantum Error Correction" section below, we detail an exemplary model for fault-tolerant quantum computing, and related technical advantages are discussed in the "Technical Advantages" section below.

[0015] System Configuration Overview

[0040] In some embodiments, an optical quantum computing system configuration includes three modules collectively configured to generate computational resource states in two spatial and one temporal dimensions using a two-dimensional circuit on an integrated photonic chip. Each resource state includes one or more clusters of encoded qubits (e.g., GKP qubits), magic states, and CV nodes stitched into hybrid "cluster states." As used herein, "stitching" refers to the creation / attachment of entanglement between different states. For example, during operation of the two-dimensional circuit, non-deterministically generated encoded qubits and magic states can be stitched to random but known subsets of sites (by generating them at some random subsets of locations and not at others), with the remaining sites occupied by deterministically generated squeezed vacuum states. An indication of whether a location is within a subset of sites is stored in a multiplexer. The encoded qubits carry quantum information and are used for CV error correction. The magic states are used to implement non-Clifford operations as desired, and Clifford operations are performed using the CV nodes of the cluster states if nearby CV nodes are available. As used herein, a "magic state" refers to a state that, when used with operations from the Clifford group that have elements that affect the permutation of the Pauli operators, facilitates universal quantum computation. For example, the eigenstates of the π / 4 gate are magic states.

[0016]

[0041] In some embodiments, hybrid cluster state generation is performed using three modules: a state preparation module, a multiplexing module, and a main computation module. The state preparation module is configured (e.g., programmed or hardwired) to generate boson qubits and magic states. The multiplexing module is configured (e.g., programmed or hardwired) to perform multiplexing of boson qubits to increase qubit generation speed and replace empty modes (e.g., failed multiplexed qubit generation) with squeezed vacuum states. As used herein, "multiplexing" refers to using multiple non-deterministic qubit generating devices in parallel and routing qubits generated by any of these devices to an output. The probability of success of at least one qubit generating device among the multiple qubit generating devices is increased (i.e., "boosted") compared to the probability of success of a single (non-multiplexed) device. The main computation module is configured (e.g., programmed or hardwired) to stitch (or "entangle") the hybrid resource states for universal fault-tolerant quantum computation and perform reconfigurable measurements on the generated resource states to complete the computation. These steps are described in more detail below.

[0017]

[0042] 1 , an exemplary system 100 for generating hybrid cluster states, according to one embodiment, includes a state factory 102, time stitching 104, spatial stitching 106, an optical quantum processing unit (QPU) 108, and a QPU controller 112. State factory 102, time stitching 104, spatial stitching 106, photonic QPU 108, and QPU controller 112 each represent logical functions that can be implemented using hardware, software, or a combination thereof. As used herein, an “active” system implementation (e.g., of system 100) refers to an implementation that includes an in-line squeezer / performs in-line squeezing and thus uses additional squeezed states and homodyne measurements, and a “passive” system implementation (e.g., of system 100) refers to an implementation that does not include an in-line squeezer / perform in-line squeezing and / or performs stitching with a beam splitter.

[0018]

[0043] The state factory 102 is operably coupled to a time stitcher 104, which is operably coupled to a spatial stitcher 106, which is operably coupled to a photonic QPU 108. Components of the state factory 102 generate GKP states and output the GKP states to the time stitcher 104, which implements a delay line loop and stitches together qubits as they are received. Components of the spatial stitcher 106 multiplex the output from the time stitcher 104 in the spatial domain to generate multidimensional hybrid resource states. The photonic QPU 108 is controlled by a QPU controller 112 to entangle hybrid resource states from multiple hybrid resource states into high-dimensional cluster states, including states for fault-tolerant quantum computation, as discussed herein. The QPU controller 112 can receive quadrature readouts 111 from the photonic QPU 108 and can send phase updates 109 to the photonic QPU 108. QPU controller 112 may also receive instructions 115 associated with program 116 and may output results 114 .

[0019] Module System Configuration Creating boson qubits using a multiple Gaussian boson sampling ("GBS") device

[0044] The generation of non-Gaussian states of light, including single-boson qubit states, has been proposed, analyzed, and developed by researchers. While high-fidelity state generation from a single GBS device is nondeterministic, GBS devices can be multiplexed to obtain fast, high-fidelity state generation, using greater hardware resources to increase the speed and fidelity of the generated states. Multiplexing GBS devices can be used to exploit the non-Gaussian resources of photon-number-resolving detectors (PNRs) and generate arbitrary logical single-qubit states for boson encoding, such as GKP and CAT-based states. Figures 2A-2C show an example of such multiplexed state generation.

[0020]

[0045] Figure 2A illustrates a single integrated photonic device implementing Gaussian Boson Sampling ("GBS")-based preparation of non-Gaussian states. In Figure 2A, light emitted from one output port is detected by a PNR detector (the "D"-shaped object in Figure 2A-2B) connected to the remaining output port, producing the correct click pattern {n i} is in a selected non-Gaussian state. The double lines represent classical (i.e., non-quantum) logic used to trigger the switches at the emission ports. Figure 2B illustrates a simplified representation of a single GBS device, and Figure 2C illustrates a state preparation device including GBS devices multiplexed in the spatial and / or time domains using classical logic, according to one embodiment.

[0021] Time-domain generation of 1D clusters

[0046] According to some embodiments of the present disclosure, cluster states are generated in one time dimension (“1D”) using one or more optical delay lines and a source that generates either a GKP qubit or a squeezed vacuum state. FIG. 3A (described below) shows an example of a setup for generating 1D cluster states. Alternatively, or additionally, cluster states can be generated using other boson qubits (e.g., including the use of a photonic controlled Z (“CZ”) gate). A CZ gate is a two-qubit gate that performs two-qubit operations. The truth table for a CZ gate is as follows:

[0022] [Table 1]

[0023]

[0047] In some embodiments, the integrated photonic chip circuit receives as input light emitted from a light source (e.g., the integrated photonic device of Figures 2A-2C) and generated using GBS state preparation. More specifically, the emitted mode is: |+> GKP(if the multiplexer is successful), or in a momentum squeezed state (if the multiplexer is |+> GKP The first mode can be either a circulating light or a non-circulating light (injected when no circulating light is generated). The first mode is swapped into an optical delay line using an interferometer (the operation of which is illustrated and described below with reference to Figure 3B), with its length set equal to the distance between subsequent optical pulses. This first mode then returns to the interferometer and interacts with subsequent modes at the CZ gate implemented by the interferometer (see Figure 3B). This interaction is repeated for each incoming mode. The effective optical circuit can be more easily visualized as an equivalent spatial representation shown in Figure 4. During the final step of operation, the circulating light is expelled from the delay line by implementing the swap using an interferometer. Thus, a one-dimensional GKP cluster state is generated. A complete exemplary device for generating one-dimensional clusters is shown in Figure 3C.

[0024]

[0048] 3A-3B illustrate the generation of a 1D qubit cluster in the time domain, according to one embodiment. On the left side of FIG. 3A, a source 300 comprising multiplexed GBS devices 302A-302D is used to generate a pulse sequence, with each pulse being in the |+> state of a selected number of selected qubits. Additional details regarding GBS devices can be found, by way of example, in U.S. patent application Ser. No. 16 / 997,601, filed August 19, 2020, entitled "Apparatus and Method for Generating Gaussian States from Non-Gaussian States," and "Conversion of Gaussian States to Non-Gaussian States Using Photon-Number-Resolving Detectors," Phys. Rev. A, 100 (2019), the entire contents of each of which are incorporated herein by reference.

[0025]

[0049] The first qubit is sent into the loop using a swap operation (top right in Figure 3B). The second qubit follows the first qubit and interacts with it with a CZ gate (bottom right in Figure 3B). The output pulse is in a 1D qubit cluster state in the time domain. The interferometer 310 in Figure 3A can function as either a swap gate or a CZ gate, depending on the settings used. Turning off the squeezer (denoted "Sq" in Figures 3A-3B) and setting the phase shifter (denoted "PS" in Figures 3A-3B) to π enables a fully reflective Mach-Zehnder interferometer (top right in Figure 3B). When the phase shifter is off and the squeezer is on, the CZ gate (bottom right in Figure 3B) is enabled. The generation of an N-site 1D cluster as described in Figure 3A involves performing a swap during the first clock cycle and performing a CZ during the remaining clock cycles except for the last clock cycle, which performs a swap (as in the first clock cycle) to kick the light out of the loop. Figure 3C is a simplified representation of the 1D qubit cluster generator of Figure 3A.

[0026] 2+1 dimensional GKP cluster

[0050] In some embodiments, the one-dimensional hybrid time cluster states described above can be stitched into higher-dimensional cluster states, including states that can be used to perform fault-tolerant quantum computations. The creation of a higher-dimensional hybrid lattice can be achieved, for example, by the interaction of pulses from multiple one-dimensional hybrid cluster generators with CZ gates. In one embodiment, a higher-dimensional hybrid lattice is created using one or more CZ gates operating on a spatial lattice of 1D clusters and qubit sources.

[0027]

[0051] In some embodiments, a 1D chain of multiple cluster generators is used to generate a (1+1)D (i.e., 2D) cubic cluster state in one spatial dimension and one time dimension, as shown in Figure 5 A. The generated 2D cubic cluster state can be stored in one or more delay lines, for example, to allow time for feedforward-based measurements, especially when post-selected gates are used.

[0028]

[0052] In other embodiments, a 2D chip can be used to generate 3D cubic cluster states in two spatial dimensions and one time dimension, as shown in Figures 5B and 5C. More specifically, Figure 5A illustrates the generation of a (1+1)D cluster state using multiple 1D cluster generators. A qubit interacting via a CZ interaction with another qubit arriving at the same time produces a (1+1)D cluster state in a cubic lattice. Figure 5B is an exemplary 2D chip layout for a (2+1)D cluster generator, according to one embodiment. The 2D chip layout of Figure 5B is similar to the 1D chip, except that the sources are arranged as a 2D lattice on the chip. Figure 5C illustrates a 3D cubic lattice generated using a 2D chip with the layout of Figure 5B, according to one embodiment. As can be seen in Figure 5C, There are two types of points that reflect the GKP|+> state (black points) and the momentum-squeezed state (white points). In some implementations, other nodes may also be in the magic state, depending on the exact cluster used.

[0029] Generation of hybrid Raussendorf-Harrington-Goyal ("RHG") lattices

[0053] In accordance with one embodiment, the generation of an exemplary cluster, i.e., a Raussendorf lattice (as a paradigmatic example), is discussed below. Instead of, or in addition to, a Raussendorf lattice, one or more other lattices useful for fault-tolerant quantum computing (e.g., non-lobed lattices) can also be selected depending on the choice of quantum error-correcting code and generated using the scheme described herein.

[0030]

[0054] FIG. 6A shows an exemplary representation of two layers of a Raussendorf lattice separately, according to one embodiment. More specifically, FIG. 6A shows alternating even and odd layers of a Raussendorf lattice. Dots represent individual computational qubits, and connections between the dots represent entanglement. The left subdiagram of FIG. 6A represents an even layer of the Raussendorf lattice, and the right subdiagram of FIG. 6A represents an odd layer of the Raussendorf lattice. FIG. 6B is a combined representation of the two layers of the Raussendorf lattice of FIG. 6A. Dots represented by "A" represent one type of qubit present in each layer and connected to corresponding qubits in the next and previous layers. Dots (qubits) represented by "B" and connections represented by "b" exist only in odd layers. Dots (qubits) represented by "C" and connections represented by "c" exist only in odd and even layers.

[0031]

[0055] FIG. 6C shows an example chip layout including multiple controlled Z (“CZ”) gates (lines connecting pairs of black dots / nodes) for generating the Raussendorf lattice of FIG. 6A. The chip of FIG. 6C can include, for example, the photonic QPU 108 of FIG. 1. The layout of FIG. 6C includes two types of sources: a qubit source (such as that shown in FIG. 2C) and a 1D cluster source (such as that shown in FIG. 3C). During operation of the chip of FIG. 6C, qubits emitted from the qubit source are entangled only with their spatially adjacent qubits, i.e., qubits emitted simultaneously. Qubits emitted from the cluster source are entangled not only with two spatially adjacent qubits, but also with modes emitted before and after them. By turning half of the qubit sources on and off at alternating time steps, a Raussendorf lattice can be generated. This lattice can serve as a resource for performing fault-tolerant quantum computation, as discussed further below.

[0032]

[0056] As shown in Figure 6C, three types of sources are integrated on the chip. The boxes represented by "D" are hybrid 1D cluster state sources, in which a sequence of entangled qubits is emitted with a time delay τ. The boxes represented by "H" are hybrid qubit sources. The dots represented by "B" and "C" are qubit sources that only emit with a time delay of 2τ, with the "B" source only emitting at (2n-1)τ and the "C" source only emitting at (2n)τ. The lines on the chip in Figure 6C represent CZ gates. The CZ gates represented by "b" are only turned on at odd times, and the CZ gates represented by "c" are only turned on at even times. These qubits and CZ gates together create the various layers of the Raussendorf lattice and the connections between those layers.

[0033] Passive version of the system configuration

[0057] Robust and stable optical quantum information can be generated by combining GKP qubits with qubit quantum error-correcting codes implemented by measurement-based quantum computation (MBQC) in hybrid continuous-variable (CV) and discrete-variable (DV) architectures. However, most well-known architectures of this type still suffer from significant challenges in that in-line squeezing in circuit-based or measurement-based implementations of CZ gates introduces noise. Furthermore, the use of deterministic GKP sources can result in high computational costs associated with multiplexing, and the demand for rapid reconfiguration in linear optical networks can impose significant loads on integrated circuits. Each of the aforementioned factors further increases the number of optical components each photon encounters as it passes through the system, thereby exacerbating losses. Losses are the most pernicious defect in optical quantum computers.

[0034]

[0058] According to some embodiments described herein, the output of a GKP qubit stochastic source can be entangled into fault-tolerant resource states for MBQC without using in-line squeezing and / or reconfigurable linear optics. The disclosed architecture can generate a three-dimensional macro-node lattice structure in one time dimension and two spatial dimensions, with each site in the lattice structure containing (or, in some implementations, consisting of) four modes. In some implementations, the generation circuitry may consist of only a single-mode source, a four-depth static circuit of balanced beam splitters, a single time-step delay line, and a homodyne detector. The generated resource states can be used similarly to CV / DV hybridized RHG cluster states, although the process is generalizable to other qubit codes. Furthermore, the symmetry of the generation circuitry allows both finite squeezing noise and uniform photon loss across the entire beam splitter network to be equivalent to local Gaussian noise in front of each detector.

[0035]

[0059] The logical error rate of the outer (qubit) code was calculated for various levels of finite squeezing and photon loss across various failure probabilities of GKP state generation. If the source fails to generate a GKP state, a squeezed vacuum state is assumed to be generated. For example, with 15 dB of squeezing and no loss, the disclosed architecture was found to tolerate a GKP failure rate of over 50%, significantly reducing the size of the per-node state preparation module and multiplexer of known systems. Furthermore, under conditions of deterministic GKP state generation, a squeezing threshold of approximately 10 dB was found (lower than that found in known systems), even though known systems ignore noise from in-line squeezing in CZ gates. The trade-off between tolerable finite squeezing noise and uniform photon loss rate for a given GKP failure rate is discussed below.

[0036]

[0060] Qubits can be encoded into optical boson modes by GKP encoding, with ideal logical 0 and 1 code words defined as follows:

number

[0037]

[0061] where μ is a placeholder for the value 0 or 1,

number

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number

number

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number

[0038]

[0062] The effect of finite squeezing can be modeled by applying an additive Gaussian boson channel to the ideal |0>p and |φ> states.

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[0039]

[0063] 50:50 beam splitter

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[0040] 3D Hybrid Macronode Architecture

[0064] A constant-depth generation circuit has been proposed for RHG lattice states that is compatible with stochastic GKP state sources. However, this proposal remains experimentally challenging because it involves the use of inline squeezing (as present in CZ gates) and time-varying circuits (i.e., different gate arrangements for even and odd time steps). Each of the aforementioned issues can be circumvented by replacing the RHG lattice target state with a computationally equivalent macro-node cluster state, where each node has multiple modes that are subject to multimode measurements.

[0041]

[0065] Figure 12 shows the main unit cell of a 3D hybrid pair cluster state in inset (A), and insets (B)-(C) illustrate steps for its creation, according to some embodiments. Figure 12 insets (B)-(D) are shown as cross sections of stacked waveguide layers in the Z direction, which coincides with the propagation direction of light through the waveguide. The 3D lattice exists in two spatial (X, Y) dimensions and one time dimension. The time dimension is divided into discrete time bins of width ΔT. Colors (represented by "G" for green, "B" for blue, "Y" for yellow, "R" for red, and "K" for black) are included in Figure 12 to indicate the relationship between source and final states. Yellow and blue indicate even- or odd-time signatures. The macronodes above the blue shapes correspond to the blue macronodes in inset (A). The macronodes above the yellow shapes correspond to the yellow macronodes in (A). The macronodes above the green shapes correspond to the green regions (i.e., partially blue, partially yellow) in inset (A). The red arrows (represented by "R") represent spatial connections, and the black arrows (represented by "K") represent temporal connections. Inset (B) of Figure 12 shows the waveguide configuration in the first layer, where each node receives input from a source in ΔT-wide time bins. The time bins of the black circle nodes are offset by ΔT / 2 relative to the white circle nodes. As indicated by the arrows, a 50:50 beam splitter is applied between pairs of modes, which generate entangled pairs (see equation (7) below). The beam splitter indicated by the black arrow creates entangled pairs connecting the states in the Z direction. Inset (C) of Figure 12, the X indicates the application of a ΔT / 2 time delay line, and the diagonal lines indicate the application of a π / 2 phase delay. Inset (D) of Figure 12, states are connected to macronode cluster states by applying four additional beam splitters between the four modes that make up each macronode. The beam splitters shown with dotted lines are applied after the beam splitters shown with solid lines. Note that the time delay lines change the time signature of a particular node.

[0042]

[0066] In some embodiments, the basic building block of a 3D hybrid pair cluster state is a type of two-mode entangled state, which can be generated by first generating a mode pair that is either a GKP or a momentum-squeezed vacuum, and then sending the mode pair through a 50:50 beam splitter. Although the constituent modes are only coupled by the beam splitter, the resulting pair corresponds to a two-mode cluster state, as revealed by the identities shown in Figure 13 (Equations 3-6), where: |φ> can be any state. From these identities, both |ψ> and |φ>>

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[0043]

[0067] In some implementations, the entangled pairs are arranged in a 3D configuration, as shown in inset (A) of Figure 12. To achieve this, at regular intervals, swap emits |φ> with probability p swapWe start with a 2D array of sources emitting momentum-squeezed states with the desired probability p swap We assume that can arise from multiplexing multiple GBS sources, with each source being an effective source. We can specify that each source generates an input mode every time step of length ΔT, but the timing of half the sources can be offset by ΔT / 2 according to their position in the 2D layout in Figure 12, inset (B). The beam splitters, delay lines, and phase delays in Figure 12, insets (B) and (C), generate the desired arrangement of pair states in (2 + 1) dimensions.

[0044]

[0068] To create a fully connected 3D resource state, four 50:50 beam splitters can be applied within each macronode, similar to a quad-rail lattice structure, as shown in inset (D) of Figure 12. A detailed pictorial representation of the resulting state is described in the "Entanglement Structure" section below. Each mode is then sent to a homodyne detector.

[0045] Equivalence with canonical hybrid cluster states

[0069] In some embodiments, a hybrid RHG cluster state is used as the canonical RHG lattice state, since there is one mode per node and its generation involves a CZ gate. The state generated by the circuit in Figure 12 is a macro-node version of this state. The case where we always measure three modes, called satellite modes, is

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[0046]

[0070] Inset (A) of Figure 15 shows a circuit representation of the beam splitter network associated with a single macronode 0 when the central mode is the top wire. Also shown is the connectivity to adjacent macronodes through the beam splitters. The circuit rules are given in the legend of Figure 15. The last four beam splitters correspond to the beam splitters in inset (D) of Figure 12. Inset (B) of Figure 15 shows the equivalent circuit to inset (A), which is derived by application of the identities of Figure 14. X0 represents the displacement X((m2 + m3 / 2). S is a squeezing gate, whose effect is to rescale the homodyne result. Inset (C) of Figure 12 shows the equivalent circuit to inset (B), which is derived from the circuit identities that shift the CZ gate towards the measurement. CX † The gates are drawn with dotted lines because they trivially affect the circuit inputs. The displacements Z1;...,4 depend on the measurements of satellite modes at adjacent macronodes.

[0047]

[0071] To economize on the description of the post-measurement state, the central mode at each macronode can be selected from the wires with inputs prepared in the GKP state whenever possible. Representing state generation and measurement by quantum circuits leads to further economization if the central mode is considered to be the top wire shown in inset (A) of Figure 15. This is because other cases can be made equivalent to this one by substituting the measurement basis at the end. Using equations (3) and (7), the beam splitter can be replaced by a CX(†) gate and squeezer. On the measurement side, the gates X(a), S(ξ), and CX jk Commutation relations between and identities for homodyne measurements <m| q X(a)= <m-a| q , <m| q S(ξ)∝ <m / ξ| q , and

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[0048]

[0072] The central mode is the encoded GKP state

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[0049] Noise Model

[0073] Any single-mode Gaussian boson channel ε that preserves the phase-space average of the vacuum state satisfies

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[0050]

[0074] Furthermore, if ε is also isotropic with respect to the phase space quadrature, then: R(θ)(ε(·))R † (θ)=ε(R(θ))(·)R † (θ) (10)

[0051]

[0075] From these identities, we can combine and exchange the uniform photon losses occurring just before the beam splitter layer in insets (B-D) of Figure 12 to act just before the homodyne detector layer in inset (A) of Figure 15. η represents the total transmission coefficient of the cumulative losses acting before each detector.

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[0052]

[0076] Once the photon loss and finite squeezing noise are accounted for as Gaussian random noise in the measured data, the reduction to the canonical RHG lattice state described above can be applied. However, reinterpreting this noisy measured data to restore the conditional displacement (a.k.a., by-product operator) at the central mode in inset (C) of Figure 15 further distorts the homodyne result for the central mode.

[0053] Threshold calculation

[0077] The correctable region for the macronode resource state can be found by Monte Carlo simulation, and each trial includes three steps: simulating the full macronode RHG lattice prepared in Figure 12, reducing it to a canonical lattice, and performing error correction on the reduced lattice.

[0054]

[0078] The noisy homodyne result of a macronode lattice can be obtained by first sampling the (ideal) quadrature phases, applying an entanglement gate, and then integrating them into the covariance matrix

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[0055]

[0079] When all modes are in GKP state, the swap-out probability p swap The threshold for is found to be 10.1 dB. With the additional restriction that every macro node has exactly one GKP state, the threshold becomes 13.6 dB. There is a significant improvement in the swap-out tolerance of the passive architecture, about 71%, as opposed to about 24% for some known active architectures.

[0056]

[0080] Without wishing to be bound by theory, we hypothesize two main reasons for the observed improvement. First, swapping a GKP mode with a momentum-squeezed state introduces noise that is correlated between its neighbors. Analysis reveals that the reduced lattice has a valid momentum-squeezed state only if all four modes at the macronode before the reduction are swapped. Therefore, the redundancy of the macronode lattice makes it more tolerant to swap-outs. Second, the by-product operators, which are conditional on measurements of neighboring GKP states, are binned and therefore do not propagate Gaussian noise. Indeed, every additional GKP state present at a given macronode provides an additional degree of local GKP error correction.

[0057]

[0081] Previous work has shown that quantum error correction (in the form of topologically protected cluster states) can be used for optical quantum computing with a probabilistic source of GKP qubits, provided the available squeezing is sufficiently high. However, such work assumes systems with both in-line squeezing and time-varying beam splitters, both of which are difficult to implement at desired noise levels. By using static linear optical circuits to generate macronode lattices, the architectures described herein according to some embodiments circumvent such obstacles, thereby making it feasible to implement topological error correction at desired (i.e., optimally low) noise levels.

[0058]

[0082] In some embodiments, the systems described herein for generating hybrid cluster states do not use experimentally demanding CZ gates. While previously considered ideal by researchers, such CZ gates have been shown to substantially degrade state quality. The cause of such degradation is believed to be inline squeezing. The embodiments described herein avoid such degradation by using circuit identities to shift all squeezing to the circuit's input (where it can be absorbed into state preparation) or output (where it manifests as conventional processing of homodyne measurement results). Furthermore, by doubling the number of modes at sites with Z-direction connectivity, the need for reconfigurable optical elements in the cluster state generation circuitry can be eliminated. The only remaining reconfigurable components are the multiplexed sources of the individual GKP states (if switches are used) and the local oscillator phase of each homodyne detector.

[0059]

[0083] By exploiting the symmetry of the resource generation circuitry, both uniform loss effects and finite squeezing effects can be integrated into the complex Gaussian noise associated with each detector. Such a model is shown to be able to treat finite squeezing noise and photon loss equally, facilitating unprecedented noise reduction.

[0060]

[0084] The circuit identity represents the built-in redundancy provided by the satellite mode of resource states described herein, arising from the permutation symmetry of the generating circuit. Having multiple GKP states per macronode is equivalent to an additional round of GKP error correction, keeping the threshold at approximately 10 dB for the full GKP case. However, providing even a single GKP state at a macronode means that the encoding state at each site behaves like a GKP state, providing significantly higher tolerance to swap-outs. At 15 dB, the highest currently reported level of optical squeezing observed (for a squeezed vacuum in bulk optics), the embodiments described herein can accommodate the replacement of more than half of that GKP state by a momentum-squeezed state. This means that the increase in the number of cluster modes is balanced by a corresponding reduction in the number of probabilistic state sources at each node, significantly easing the multiplexing requirements. Collectively, the results achievable by the disclosed system substantially facilitate the realization of fault-tolerant and scalable optical quantum computers.

[0061] Entanglement structure

[0085] The relationship between the generation circuit and the entanglement structure of the generated state is described below. After the state generation stage shown in insets (C) and (D) of Figure 12, the array of modes becomes as shown in inset (A) of Figure 16. Figure 16 shows a graph of the hybrid macronode 3D cluster state. Inset (A) of Figure 16 shows the 2D mode layout, with ΔT / 2 offset modes consistent with insets (C) and (D) of Figure 12. That is, the modes at the nodes marked with black circles are offset in time by ΔT / 2 relative to the modes at the nodes marked with white circles. Each macronode consists of four modes, denoted 1 through 4. Inset (B) of Figure 16 shows the 3D arrangement of the four-mode macronode. For clarity, labels labeled A–F have the color corresponding to the given letter, with deeper layers being lighter in color (green in the X and Y directions, red in the Z direction; light green is represented by “LG,” dark green by “DG,” light red by “LR,” and dark red by “DR”). Five modes corresponding to the back faces of the cube have been omitted from the unit cell. The connectivity in the XY plane is identical to that of the front face. Figure 16 inset (C) shows the macronode graph edges for each bond in inset (B). The top six configurations correspond to CZ gates with weight 1 and connect pairs of modes as in inset (A) of Figure 12. The bottom six configurations correspond to CZ gates with weights of ±1 / 4 (darker edges are positive and lighter edges are negative) and show the mode connectivity after the stage shown in inset (D) of Figure 12.

[0062]

[0086] Recall that the solid circle nodes in Figure 12 indicate that these lattice sites exist in time modes and are offset by ΔT / 2 compared to the lattice sites of the open circle nodes. The grouping of modes into macronodes is indicated by yellow (blue) squares and rectangles, which also indicate that these macronodes are offset by ΔT / 2. When the resource is built up to a point between stages C and D in Figure 12, it corresponds to projected entangled pair states (PEPS) for the CV / DV RHG cluster state, as shown in inset (A) of Figure 12. The precise identification of the waveguide modes by the graph nodes is shown in Figure 16.

[0063]

[0087] As described herein, a 4-to-1 reduction of modes per macronode can correspond to the application of a projection operator equivalent to performing four beam splitters and three homodyne measurements. Similarly, other measurements can be performed after the beam splitter (leading to more general operations than can be realized with a canonical lattice). A four-layer graph for the state after the beam splitter but before the homodyne measurements is shown in the inset (C) of Figure 16. The dark / light edge coloring is consistent with the plus / minus signs in the real-valued adjacency matrix of the state, which results from a graphical calculation for a Gaussian pure state when all modes are initially in a squeezed state.

[0064]

[0088] In some implementations of the passive system configurations described above, the passive grid is a dimensional 2 N (e.g., 2, 4, 8, 16) lattices. Additional details regarding passive implementations are described, for example, in “Fault-Tolerant Quantum Computations with Static Linear Optics,” I. Tzitrin, et al., available at https: / / arxiv.org / abs / 2104.03241, the entire contents of which are incorporated herein by reference for all purposes.

[0065]

[0089] As mentioned above, Figures 6A-6C illustrate how a Raussendorf lattice can be generated, according to one embodiment. Each node represents the output of a single-mode source (emitting either a GKP-plus state or a momentum-squeezed state at each time step), and each link / edge represents the application of a CZ gate (e.g., implemented using a pair of squeezers sandwiched between two beam splitters). However, in some instances, the use of a CZ gate in the generation procedure can be architecturally challenging, for example, when the active squeezing element is implemented using measurement-based squeezing and a feedforward gate. Thus, in some embodiments, all of the CZ gates represented by links / edges in FIGS. 6A-6C are replaced with 50:50 beam splitters (e.g., 50:50 beam splitters may be easier to implement experimentally, since measurement-based squeezing and feedforward gates may require additional high-speed electronics on the photonics chip and may inject unwanted noise into the light), and each node in FIG. 6B is modified as follows: the "B" and "C" nodes are modified to include four separate sources of squeezed light / GKP-plus states routed through four additional beam splitters; the "A" node is modified to include six separate sources of squeezed light / GKP-plus states routed through six additional beam splitters. An example of such a modification is shown in FIG. 7, and an example of the entire associated passive system configuration chip layout is shown in FIG. 8.

[0066]

[0090] In Figure 7, each link / edge represents a 50:50 beam splitter (rather than a CZ gate). Four different types of nodes are shown in Figure 7, each configured to emit either four modes or six modes. The representation in Figure 7 is similar to Figure 6B. "A" nodes (and "a" links / edges) are implemented in each layer of the state generator, "B" nodes and "b" links / edges are implemented in odd-numbered layers of the state generator, and "C" nodes and "c" links / edges are implemented in even-numbered layers of the state generator. The arrows in each panel of Figure 7 indicate that the interaction of the rightmost beam splitter is implemented next. The double lines in the rightmost diagram of Figure 7 indicate beam splitters that cannot be implemented simultaneously with the solid-line beam splitters, but the order of the solid / double-line beam splitters is interchangeable in the rightmost diagram. In other words, any particular macronode (i.e., the set of four "B" and / or "C" modes in Figure 7) is occupied by only four states. The generated states contain more modes and have more complex entanglement structures, but can be reduced to a Raussendorf-Harrington-Goyal ("RHG") lattice based on local homodyne measurements.

[0067]

[0091] As mentioned above, Figure 8 is a schematic diagram of a passive system configuration chip layout for generating states, including nodes such as those shown in Figure 6, according to one embodiment. In Figure 8, one mode in each of the mode pairs "P" is delayed in time by Δt to connect layers at different time steps (also shown in Figure 10B). In each mode pair in Figure 8,

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[0068]

[0092] A legend for the components of the circuit of Figure 8 is shown and described with reference to Figures 10A-10B. Figure 10A shows the dumbbell-shaped entangled states of Figure 8, each containing two modes interacting at the beam splitter. Depending on the choice of the two modes used (small circles at the ends of each dumbbell-shaped entangled state), a variety of different entangled states can be implemented, examples of which are listed in Figure 11. Figure 10B shows the

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[0069]

[0093] Each entangled state shown in Figure 8 can be one of at least three types: (a) both modes are the appropriate squeezed state; (b) one mode is a squeezed state and one mode is the selected GKP state; or (c) both modes are the selected GKP state. Figures 11A-11C show equivalent circuits for each of these different types of entangled state, respectively, according to one embodiment.

[0070] Generalizing passive architectures - passive construction of general graph states

[0094] In some embodiments, a general algorithm is defined to take a set of N Bell pairs and, via interferometry and subsequent destructive measurements, combine half of those N Bell pairs into a single vertex, with N edges providing the other half of each pair. The single unit in question can be written as a graph transformation shown in Figure 9, where circle "C" indicates the mode that interacts with the interferometer.

[0071]

[0095] Some known beam splitter networks use four splitters to construct cluster states. Such methods can be extended to eight-mode macronodes by repeating the procedure with another four modes, adding another series of beam splitters that connect mode 1 to mode 1, mode 2 to mode 2, and so on. This pattern can be extended to eight-mode macronodes of size 2. NThis can continue to scale to a generalized beam splitter network that can entangle macronodes of 4 modes. To accommodate codes of various sizes, additional auxiliary states can be used to artificially increase the size of the macronodes, and then these additional modes can be removed by position measurements. Additional details regarding the construction of cluster states using 4 splitters are described, for example, in “Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer,” by Bourassa, EJ, et al., Quantum (5), February 2021, available at https: / / arxiv.org / abs / 2104.03241, and in “Fault-Tolerant Quantum Computation with Static Linear Optics,” by Tzitrin, I., et al., the entire disclosures of each of which are incorporated herein by reference.

[0072]

[0096] In some embodiments, the algorithm implements macro node construction from base graphs of any shape and any size. For example, in one implementation, the algorithm may be configured to implement a hypercube state, while in another implementation, the algorithm may be configured to implement a non-hypercube state.

[0073]

[0097] Known interferometers based on the Discrete Fourier Transform ("DFT") can introduce complex weighted edges into the graph between the external modes of Figure 9, causing the output graph states to no longer be true cluster states. Known universal N-mode multiport interferometer methods contain a significant amount of redundancy, which can be undesirable. Minimizing the number of optical elements used in an optical network can reduce loss and noise within the network.

[0074]

[0098] In some implementations, the minimum criterion for a unitary implementing the graph transformation shown in FIG. 9 that contains the desired output and minimizes the required optical elements is that one row or column of entries

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[0075]

[0099] A unitary matrix with the structure of equation (11) can be constructed using a beam splitter network.

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[0076]

[0100] The subscripts represent the modes in which the operators act, and the products follow the order of the operators from left to right (i=1 to i=N-1). This network uses only N-1 beam splitters.

[0077]

[0101] In some implementations of the configurations described in this section, the passive grid can be determined for grids of dimensions 6, 10, 12, 14, 18, etc.

[0078] quantum error correction

[0102] The following provides examples of applying error correction procedures to computations in the presence of errors, according to some embodiments. In the first example, finite squeezing errors are addressed and probabilistic code state generation is considered. In Method 1 below, steps for performing fault-tolerant computations are presented, according to some embodiments. Method 1 receives as input a specific computational task executed on a quantum computer and specified by a user. The computational task subroutine includes implementations of quantum error correction rounds interleaved with implementations of logical operations. The quantum error correction procedure is further detailed in Method 2. The quantum error correction procedure receives as input measurement results implemented to perform logical operations and outputs reliable syndrome data. As used herein, "syndrome data" refers to the results of measurements performed to determine whether an error has occurred and, if so, which sites have the error. "Reliable" refers to results obtained by performing such measurements multiple times and polling the results to reduce the sensitivity of the results to measurement imperfections. The error correction procedure includes the use of a decoder to process the measurement data. An example of a dual decoder including an inner (boson) decoder and an outer (qubit) decoder is described in methods 3 and 4 below.

[0079] Method 1: Procedure for performing fault-tolerant quantum computation Input: Input to computational tasks and problems 1. Compilation: Based on the computational task and input to the problem, determine the appropriate logic circuit containing layers of operations and select the appropriate quantum error-correcting code and decoder. 2. State initialization. Generate cluster states based on the selected quantum error correcting code and logic circuit. An example of a generated state can include a GKP qubit in a known subset of modes and a squeezed vacuum state in the remaining modes. 3. Implementing logic circuits. Logic circuits are used to repeatedly apply layers of logical operations / measurements and perform error correction. (a) Perform one layer of logical operations by measuring a subset of sites in an appropriate (e.g., predetermined) basis as appropriate for the quantum computation, the selected quantum error-correcting code, and the measurement results from the previous error-correction and logical operation round. The measurements can include, for example, homodyne measurements in optical modes. (b) Perform error correction using Method 2 on the measurement results obtained in the previous step. 4. Processing of Measurement Results: The logical measurement results are processed to obtain output for the computational task. Output: The output of a computational task given an input.

[0080] Method 2: Procedure for implementing quantum error correction Input: (1) a measurement result from performing a logical operation, e.g., a real-valued homodyne measurement result, and (2) information about which sites contain GKP qubits and squeezed vacuums. 1. Decoder Implementation: The exemplary decoder implements a two-step procedure as follows: (a) Inner Decoder: Processes the real-valued homodyne measurement results to obtain a binary result representing a bit value using the local and global information obtained by Method 3. It uses information about which sites contain which states. (b) Outer decoder and error correction: Using the additional information provided by the previous step, apply outer code error correction using Method 4. Output: reliable syndrome results.

[0081] Method 3: Exemplary Inner Decoder input:

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[0082] Method 4: Exemplary Outer Decoder: Minimum Weight Perfect Matching ("MWPM") Input: Qubit measurement results from Method 3 1. Syndrome identification: From the input qubit measurements, construct the associated stabilizer measurements. 2. Constructing the Matching Graph: Construct the complete graph using: A vertex containing a pair of unsatisfied syndromes (with additional vertices if specific boundary conditions are desired). An "unsatisfied" syndrome represents a measurement in which an error was detected. · Edges connecting every pair of vertices. · Weights assigned to edges reflecting the probability of the most likely error that resulted in an unsatisfied syndrome pair. 3. Matching method: Find the minimum weight perfect matching by running, for example, Edmonds' algorithm on the matching graph from the previous step. 4. Recovery operations: Infer one or more recovery operations from the matching graph. 5. Correction: Interpret syndrome results taking into account recovery maneuvers. Output: reliable syndrome results.

[0083] Illustrative Analysis Noise Model

[0103] In some embodiments, the cluster states generated are populated by two types of states previously described herein: GKP-encoded |+> states and momentum-squeezed states. The position wavefunction of an ideal GKP state is:

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[0084]

[0104] Alternatively, to model state initialization errors, we use the ideal GKP |+> gkp Given the state, we consider the noise channel N with probabilities 1-p0 and p0, respectively. Ygkp or N Yp The ideal GKP |+> gkp This method closely approximates real momentum states in position space, but has a periodic structure in momentum space (reviewed below).

[0085]

[0105] There are several reasons for modeling state preparation errors using the Gaussian white noise channel of equations (16) and (17). For example, many CV gates use measurement-based squeezing operations that inherently lead to imperfections modeled as Gaussian white noise channels. Furthermore, this type of noise is closely related to pure losses, and tracing a pure loss channel through an amplifier with reversed intensity results in a Gaussian white noise channel. This relationship can play an important role in situations where losses can be treated in this way, such as during measurement imperfections.

[0086]

[0106] A Gaussian white noise channel is easily described using the Heisenberg picture. Consider the quadrature operators of N modes, where

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[0087] Initializing the cluster state

[0107] As discussed in the previous subsection, a vector of length 2N (where N is the number of modes) can first be initialized to store the averages of the q and p quadratures for each mode. Initially, the average of all quadratures is 0. For each mode, a momentum-squeezed state is prepared with probability p0 and a GKP|+> state is prepared with probability (1-p0). Next, CZ gates (e.g., "perfect" or "ideal" CZ gates) are applied according to the structure of the cluster states, i.e., whenever there is an edge between two nodes in the lattice. Some of the CZ gates may be inverted to conform to the CV toric code convention.

[0088]

[0108] Recall that the symplectic transformation of the CZ gate in the (q1,q2,p1,p2) basis order is given by

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[0089]

[0109] Note that A2 is the adjacency matrix for two modes, i.e., A2 is a symmetric binary matrix whose ijth entry is 1 if the two modes are connected by an edge, and 0 otherwise. CZ † For the application of , it is estimated that A2 → -A2. (Note: the notation † for a given gate denotes its Hermitian conjugate.) Therefore, the symplectic matrix corresponding to the connection of all optical modes to cluster states is given by

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[0090]

[0110] Finally, the momentum values ​​are measured, where only the momentum component of the noise matrix given by:

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[0091] Inner Decoder

[0111] In some embodiments, the standard map from homodyne measurements to bit values ​​is a binning function derived from the translational symmetry of the original GKP state, i.e., perfect periodicity in the q and p directions. The subscript "gkp" is omitted where the context is clear. |+> and |->The state has momentum

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[0092]

[0112] To illustrate the importance of such a translator, consider the example of a momentum-squeezed state at the center of a principal plane surrounded by four GKP states. -1 The large amount of CV noise is distributed symmetrically from the q-quadrature of the momentum state to the p-quadrature of the connected GKP states. However, due to the periodicity of the GKP states, the net effect is either an identity gate or a Z-gate applied to all surrounding GKP states, which exchanges with the stabilizers in the RHG lattice.

[0093]

[0113] A translator is desirable for hybrid system configurations that take into account the encoding phase of the computation, since it is the same regardless of the subsequent logical operations of the computation. In other words, it knows which modes are GKP states and p-squeezed states, and it knows where CZ gates have been applied to form cluster states. As a numerical example shows, if the distribution of p-homodyne results is examined at this stage, the p-homodyne results are sampled from a periodic configuration of Gaussian distributions, each with a covariance Σ p and each Gaussian is a point

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[0114] Suppose we obtain a value p after a homodyne measurement. From the above discussion, the candidate distributions that could have generated p are those with covariance Σ p and the lattice points

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[0115] Therefore, the lattice point that is most likely to have produced point p is given by:

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[0096] External Decoder

[0116] After obtaining and binning the results of the homodyne measurements, error correction on the outer qubit code can be performed. The details of the error correction problem were summarized in Method 4 with respect to a particular (e.g., common) choice of decoding algorithm, namely, Minimum Weight Perfect Matching (MWPM). Note, however, that countless other decoders can be used.

[0097]

[0117] In the weight assignment step, analog CV information (full homodyne measurements) can be included along with the position of the p-squeezed states in the lattice.

[0098] Numerical example

[0118] The disclosed heuristic translator can identify directions in p-space that have significantly more noise than others by replacing some GKP states with p-squeezed states and then applying a CZ gate to encode them into cluster states corresponding to selected codes. The space orthogonal to these noisy directions has a small amount of noise. A change of basis can be performed to determine CV data along these directions with various levels of noise. These directions represent linear combinations of the original modes and, in the ideal case, still yield integer-valued results. Using appropriate binning to account for the self-consistency of the integer linear combinations, results can be determined in this basis. Reversing the change of basis returns us to an integer-valued vector for the code lattice, which, after modulating by 2, returns a bit string representing the qubit-level result.

[0099]

[0119] An example of how the heuristic translator operates for the case of a p-squeezed state surrounded by four GKP states is provided below in accordance with some embodiments, and includes assigning bit strings to five homodyne results.

[0100] Method 5: An exemplary inner decoder for one p-squeezed state surrounded by GKP states input:

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[0101]

[0120] FIG. 17 illustrates the modularity of the system configuration of a GBS generation chip, according to some embodiments. In some implementations, the GBS generation chip is reconfigurable. In other implementations, the GBS generation chip is reconfigurable but includes an integrated PNR to reduce losses. The GBS generation chips of the present disclosure may be partially or completely stored and / or operated in a cryogenic environment. For example, in some embodiments, state generation is performed in a cryostat, and subsequent calculations are performed at room temperature. Each GBS generation chip includes an active switching system with high-speed switches that may optionally not be reconfigurable. Depending on the application, some delay between the generation phase and the switching phase may be beneficial.

[0102] Technical Advantages Modularity

[0121] Various aspects of the computation described here (state preparation, multiplexing, cluster generation, and measurement) have different associated hardware specifications. These hardware specifications facilitate a modular design where different tasks can be performed on different chips. For example, the generation of boson-encoded states can be performed using non-reconfigurable circuitry with on-chip PNR. Cluster stitching can also be performed on a non-reconfigurable chip. Measurement of the generated clusters can be performed using reconfigurable homodyne detection with feedforward from measurements on other homodyne detectors.

[0103] Minimal cryogenic requirements

[0122] The state generation chips described herein can include low-loss non-reconfigurable circuitry in a static integration platform, thus allowing the use of on-chip PNR, and optionally the entire chip can be placed in a cryostat, allowing the rest of the system (e.g., the switching network for state generation) to operate at room temperature.

[0104]

[0123] Keeping the switching network at room temperature can help take advantage of delays introduced when extracting light from the cryostat. Cluster manipulation can be performed using reconfigurable homodyne detection and delay lines to enable feedforward. Thus, cluster generation and manipulation can be performed on-chip. On-chip homodyne detection can be faster than detection using superconducting detectors, thus reducing losses associated with delay lines present in the cluster manipulation stage.

[0105] Homodyne detection sets the time scale

[0124] In some embodiments, the time scale for cluster generation and cluster manipulation is set by the time scale of homodyne detection (or by any other slower process that may be present during the final generation procedure). This can be advantageous because homodyne detection can be much faster than PNR detection during the multiplexing procedure and / or threshold detectors during dual-rail encoding. Faster time scales can mean that the cluster generation delay lines can be shorter, thus resulting in lower losses. These advantages can be observed in Figures 2A, 2C, and 4C, for example, in that the delay lines, represented by τ in Figures 2A and 2C, are shorter in the system designs described herein compared to known photonic systems.

[0106]

[0125] 18 is a flow diagram illustrating a first method for generating hybrid cluster states according to some embodiments. As shown in FIG. 18 , method 1800 includes, at 1802, receiving an input vector of a homodyne measurement and a noise model. At 1804, at least one direction having a noise level above a predetermined threshold is identified based on the input vector of the homodyne measurement and the noise model. At 1806, a change of basis is performed on the input vector of the homodyne measurement (e.g., using a transformation matrix) based on the identified at least one direction to generate a first modified vector, and at 1808, a transformation is applied to the first modified vector to generate a second modified vector. The binning operation can be based on, for example, a map from the homodyne measurement results to bit values. Alternatively, or additionally, the rounding operation can be based on, for example,

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[0107]

[0126] 19 is a flow diagram illustrating a second method for generating hybrid cluster states according to some embodiments. As shown in FIG. 19, method 1900 includes receiving an input vector of a homodyne measurement at 1902 and performing a change of basis on the input vector of the homodyne measurement to generate a first modified vector at 1904. At 1906, a transform is applied to the first modified vector to generate a second modified vector. The transform may include a binning operation and a rounding operation. The binning operation may be based on, for example, a map from the homodyne measurement results to bit values. Alternatively, or additionally, the rounding operation may include:

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[0108]

[0127] While various embodiments have been described and illustrated herein, various other means and / or structures for performing the functions and / or obtaining the results and / or one or more of the advantages described herein, and each such variation and / or modification, are possible. More generally, all parameters, dimensions, materials, and configurations described herein are intended to be examples, and the actual parameters, dimensions, materials, and / or configurations will depend on the particular application or applications for which the present disclosure is used. It is understood that the foregoing embodiments are presented by way of example only, and that other embodiments may be practiced otherwise than as specifically described and claimed. Embodiments of the present disclosure are directed to each individual feature, system, article, material, kit, and / or method described herein. Furthermore, any combination of two or more of such features, systems, articles, materials, kits, and / or methods, provided such features, systems, articles, materials, kits, and / or methods are not mutually inconsistent, is within the inventive scope of the present disclosure.

[0109]

[0128] Also, various concepts may be embodied as one or more methods, examples of which are provided. Actions performed as part of a method may be ordered in any suitable manner. Thus, while the exemplary embodiments show actions as sequential, embodiments may be constructed in which actions are performed in a different order than illustrated, and may include performing some actions simultaneously.

[0110]

[0129] All definitions defined and used herein should be understood to supersede dictionary definitions, definitions within documents incorporated by reference, and / or ordinary meanings of the defined terms.

[0111]

[0130] As used herein, a "module" may be, for example, any assembly and / or set of operably coupled electrical components related to the performance of a particular function, and may include, for example, memory, a processor, electrical traces, optical connectors, software (stored and executed in hardware), etc.

[0112]

[0131] The indefinite articles "a" and "an," as used in the specification and claims, unless expressly indicated otherwise, should be understood to mean "at least one."

[0113]

[0132] The term "and / or," as used in the specification and claims, should be understood to mean "either or both" of the elements so connected, i.e., elements that are present conjunctively in some cases and disjunctively in other cases. Multiple elements listed with "and / or" should be interpreted similarly, i.e., "one or more" of the elements so connected. Other elements, whether related or unrelated to those specifically identified elements, other than the elements specifically identified by the "and / or" clause may optionally be present. Thus, as a non-limiting example, a reference to "A and / or B," when used in conjunction with open-ended language such as "comprising," can, for example, in one embodiment refer to A only (optionally including elements other than B); in another embodiment refer to B only (optionally including elements other than A); or in yet another embodiment refer to both A and B (optionally including other elements).

[0114]

[0133] As used in this specification and the claims, "or" should be understood to have the same meaning as "and / or" as defined above. For example, when separating elements in a list, "or" or "and / or" shall be interpreted as inclusive, i.e., including at least one (or more) of several elements or a list of elements, and, optionally, additional items not in the list. Only terms clearly indicated to the contrary, such as "only one of" or "exactly one of," or, when used in the claims, "consisting of," shall refer to the inclusion of exactly one element of several elements or a list of elements. In general, the term "or," when used herein, shall only be interpreted as indicating exclusive alternatives (i.e., "one or the other, but not both") when accompanied by exclusive language, such as "either," "one of," "only one of," or "exactly one of." "Consisting essentially of," when used in the claims, shall have its ordinary meaning as used within patent law.

[0115]

[0134] As used in this specification and claims, the phrase "at least one" in reference to a list of one or more elements should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of every element specifically listed in the list of elements, and not excluding any combination of elements in the list of elements. Also, by this definition, elements other than those specifically identified in the list of elements referred to by the phrase "at least one" may optionally be present, whether related or unrelated to the specifically identified elements. Thus, as a non-limiting example, "at least one of A and B" (or, equivalently, "at least one of A or B," or, equivalently, "at least one of A and / or B") can refer, for example, to: in one embodiment, at least one, optionally two or more, A, and no B (and optionally including elements other than B); in another embodiment, at least one, optionally two or more, B, and no A (and optionally including elements other than A); in yet another embodiment, at least one, optionally two or more, A, and at least one, optionally two or more, B (and optionally including other elements).

[0116]

[0135] In the claims and the above specification, all transitional phrases such as "comprise," "comprise," "possess," "have," "include," "involve," "hold," "consist," and the like, are to be understood as open-ended terms, i.e., to mean including but not limited to. Only the transitional phrases "consisting of" and "consisting essentially of" shall be closed or semi-closed transitional phrases, respectively, as set forth in the United States Patent Office Manual of Patent Examining Procedures, Section 2111.03.

Claims

1. 1. An apparatus comprising an integrated circuit, the integrated circuit comprising: a plurality of sources including a qubit source and a cluster source; a plurality of control Z gates, each control Z gate from the plurality of control Z gates connecting at least two sources from the plurality of sources; the integrated circuit is configured, during operation, to sequentially energize a subset of sources from the plurality of sources to generate a Raussendorf lattice based on interactions between sources from the plurality of sources. Device.

2. 10. The apparatus of claim 1, wherein each control Z gate from the plurality of control Z gates is a two-qubit gate.

3. The apparatus of claim 1 , wherein the cluster source is a 1D cluster source.

4. 10. The apparatus of claim 1, wherein during operation of the integrated circuit, a qubit emitted from the qubit source is entangled with a qubit emitted from at least one spatially adjacent qubit source of the qubit source.

5. a plurality of optical circuits configured to generate light having a plurality of output states by Gaussian Boson Sampling (GBS); a plurality of photon number resolving detectors (PNRs) operably coupled to the plurality of optical circuits, the plurality of photon number resolving detectors (PNRs) configured to generate qubit clusters based on the plurality of output states; a multiplexer operably coupled to the plurality of PNRs, the multiplexer configured to perform multiplexing of the qubit clusters and replace empty modes with squeezed vacuum states, thereby generating light having a plurality of hybrid resource states; an integrated circuit operably coupled to the multiplexer, the integrated circuit configured to entangle hybrid resource states from the plurality of hybrid resource states into a high-dimensional cluster state including states for fault-tolerant quantum computing; A system comprising:

6. 6. The system of claim 5, wherein the integrated circuit is configured to entangle the hybrid resource states from the plurality of hybrid resource states into the high-dimensional cluster state without using reconfigurable linear optics.

7. The system of claim 5 , wherein the high-dimensional cluster state comprises a three-dimensional macro-node lattice structure in one time dimension and two space dimensions.

8. The system of claim 5 , wherein the high-dimensional cluster state comprises a three-dimensional lattice structure, and each site from a plurality of sites of the lattice structure comprises at least four modes.

9. 1. An apparatus comprising an integrated circuit, the integrated circuit comprising: Multiple multimode sources and a plurality of beam splitters operably coupled to the plurality of multimode sources, each beam splitter from the plurality of beam splitters connecting at least two multimode sources from the plurality of multimode sources; the integrated circuit is configured, in operation, to sequentially energize a subset of multimode sources from the plurality of multimode sources and generate a Raussendorf grating based on interactions between the multimode sources from the plurality of multimode sources and at least one homodyne measurement. Device.

10. 10. The apparatus of claim 9, wherein the Raussendorf lattice includes a first layer including a set of qubits of a first type and a second layer including a set of qubits of a second type different from the first type.

11. 10. The apparatus of claim 9, wherein during operation of the integrated circuit, qubits emitted from the plurality of multi-mode sources are entangled with qubits emitted from at least one spatially adjacent multi-mode source of the plurality of multi-mode sources.

12. 12. The apparatus of claim 11, wherein the qubits emitted from the multiple multi-mode sources are further entangled with earlier emitted modes and later emitted modes.

13. a plurality of optical circuits configured to generate light having a plurality of output states by Gaussian Boson Sampling (GBS); a plurality of photon number resolving detectors (PNRs) operably coupled to the plurality of optical circuits, the plurality of photon number resolving detectors (PNRs) configured to generate qubit clusters based on the plurality of output states; a multiplexer operably coupled to the plurality of PNRs, the multiplexer configured to perform multiplexing of the qubit clusters to generate light having a plurality of hybrid resource states; an integrated circuit operably coupled to the multiplexer, the integrated circuit configured to entangle hybrid resource states from the plurality of hybrid resource states into a higher-dimensional cluster state based on interactions between hybrid resource states from the plurality of hybrid resource states and at least one homodyne measurement; A system comprising:

14. 14. The system of claim 13, wherein the integrated circuit is configured to entangle the hybrid resource states from the plurality of hybrid resource states into the high-dimensional cluster state without using reconfigurable linear optics.

15. The system of claim 13 , wherein the high-dimensional cluster state comprises a three-dimensional macro-node lattice structure in one time dimension and two space dimensions.

16. The system of claim 13 , wherein the high-dimensional cluster state comprises a three-dimensional lattice structure, and each site from a plurality of sites of the lattice structure comprises at least four modes.

17. 14. The system of claim 13, wherein the integrated circuit is configured, during operation, to sequentially energize a subset of multimode sources from a plurality of multimode sources and generate a Raussendorf grating based on interactions between the hybrid resource states from the plurality of hybrid resource states and the at least one homodyne measurement.

18. 20. The system of claim 17, wherein the Raussendorf lattice includes a first layer including a set of qubits of a first type and a second layer including a set of qubits of a second type different from the first type.

19. The system of claim 13 , wherein the light has the plurality of hybrid resource states to replace empty modes with squeezed vacuum states during operation of the integrated circuit.

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