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Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer

J. Eli Bourassa, Rafael N. Alexander, Michael Vasmer, Ashlesha Patil, Ilan Tzitrin, Takaya Matsuura, Daiqin Su, Ben Q. Baragiola, Saikat Guha, Guillaume Dauphinais, Krishna K. Sabapathy, Nicolas C. Menicucci, Ish Dhand

arXiv:2010.02905v2quant-ph

TL;DR

Scalable bosonic-qubit resources are difficult to generate and combine, motivating a hybrid architecture that uses continuous-variable resources while retaining bosonic qubits for low-noise operations. The paper introduces a tailored decoder for this architecture and finds fault-tolerant operation below swap-out thresholds, including approximately 13.3% at 15 dB squeezing.

  • Problem

    Scalable generation and combination of resources made entirely from bosonic qubits are difficult, motivating a scheme that combines continuous-variable resources with sufficiently concentrated bosonic qubits.

  • Method

    The architecture uses hybrid cluster states with bosonic qubits and squeezed-state modes, decoded through noisy measurement values and knowledge of squeezed-state locations.

  • Results

    Fault-tolerant quantum computation is possible below a swap-out probability of approximately 23.6%, while 15 dB squeezing yields a maximum tolerable swap-out probability of approximately 13.3%.

  • Takeaways & Limitations

    The hybrid design can handle state imperfections using a tailored two-tier decoder that combines continuous- and discrete-variable syndrome data.

  • Takeaways & Limitations

    The architecture relies heavily on GKP encoding, while incorporating non-Gaussian state imperfections, gate and detector noise, and transmission losses remains future work.

Abstract

from arXiv · show

Photonics is the platform of choice to build a modular, easy-to-network quantum computer operating at room temperature. However, no concrete architecture has been presented so far that exploits both the advantages of qubits encoded into states of light and the modern tools for their generation. Here we propose such a design for a scalable and fault-tolerant photonic quantum computer informed by the latest developments in theory and technology. Central to our architecture is the generation and manipulation of three-dimensional hybrid resource states comprising both bosonic qubits and squeezed vacuum states. The proposal enables exploiting state-of-the-art procedures for the non-deterministic generation of bosonic qubits combined with the strengths of continuous-variable quantum computation, namely the implementation of Clifford gates using easy-to-generate squeezed states. Moreover, the architecture is based on two-dimensional integrated photonic chips used to produce a qubit cluster state in one temporal and two spatial dimensions. By reducing the experimental challenges as compared to existing architectures and by enabling room-temperature quantum computation, our design opens the door to scalable fabrication and operation, which may allow photonics to leap-frog other platforms on the path to a quantum computer with millions of qubits.

1 Introduction

Photonics offers room-temperature, network-compatible quantum computing with flexible error-correction options, but scalable architectures face difficult resource-generation requirements. Existing proposals occupy two extremes: deterministic CV resources with demanding DV sources, or entirely bosonic-qubit resources that are difficult to generate and entangle.

  • Room-temperature photonics could support miniaturization, mass manufacturing, inexpensive components, faster operation, and rapid scaling through existing silicon technologies.
  • Photonic platforms are intrinsically compatible with communication technology, enabling high-fidelity connections between modules without noisy transduction.
  • Photonic architectures can flexibly use error-correcting codes, including mode-to-qubit encodings and high-dimensional codes using temporal degrees of freedom.
  • One architecture class uses scalable CV entangled resources for DV qubits but requires deterministic, on-demand DV resource generation.

Overview of Architecture

The architecture combines probabilistically generated GKP qubits with squeezed states in a hybrid cluster state, replacing unavailable qubits rather than erasing them. A tailored decoder handles noise from squeezed-state measurements, while planar photonic implementation supports scalable fault-tolerant operation.

  • Resource generation: GKP qubits generated by multiplexed GBS devices are probabilistic, and near-certain production otherwise requires very many devices.GBS devices use Gaussian operations and photon-counting measurements to conditionally produce GKP qubits.
  • Hybrid resource state: Failed GKP sources are replaced by squeezed vacuum states, allowing the mode to remain entangled and encode logical information in the hybrid cluster.The replacement avoids erasing failed qubits from the lattice and reduces the required number of GBS devices.
  • Scalable implementation: The architecture uses a two-dimensional integrated photonic chip to produce a cluster state in one temporal and two spatial dimensions, with constant optical depth per mode as the qubit count grows.
  • Decoding: Squeezed-state measurements inject known random noise into neighboring modes, so the decoder combines noisy measurement values with squeezed-state locations before standard qubit decoding.
  • Contribution: The design supports fault-tolerant computation with optical GKP or squeezed states at room temperature using a moderately sized planar photonic chip.
  • Contribution: The hybrid resource state, accompanying decoder, and hardware-friendly architecture are presented as the work’s main results.

2 Background on Quantum Computation Using CV Systems

CV quantum computation uses bosonic modes and cluster states, while bosonic encodings such as GKP protect discrete information against certain noise processes. The section describes Gaussian operations, GKP fault-tolerance, GBS state preparation, and concatenation with topological codes.

  • CV quantum computation: CV quantum computation operates on infinite-dimensional bosonic modes, but directly encoding data across the full Hilbert space is physically impractical because noise causes uncorrectable high-weight errors.CV Gaussian computation forms an efficiently simulable sub-theory, motivating structured bosonic encodings.
  • Bosonic qubit encodings: Bosonic qubit encodings restrict each mode to a two-dimensional subspace that can support convenient state preparation, gates, readout, and sometimes error correction.Examples include GKP, dual-rail, cat, hypercat, binomial, and rotation-symmetric codes.
  • GKP qubits: GKP encoding maps qubit Clifford operations to deterministic CV Gaussian operations, including displacements, phase-space rotations and shears, and CV entangling gates.These operations can be implemented with linear optics, homodyne detection, Gaussian states, squeezers, phase shifters, and beam splitters.
  • GKP qubits: GKP qubits provide intrinsic protection against small phase-space displacements, including weak photon loss, while larger residual errors require a second layer of qubit error correction.Displacements smaller than √π/2 can be corrected through non-destructive GKP-stabilizer measurements.
  • Error correction: The architecture uses two decoding stages: an inner decoder restores each mode to the GKP code subspace, and an outer decoder produces recovery operations from higher-level qubit-code syndromes.The inner decoder can also estimate relative likelihoods of discrete qubit-level errors.
  • Physical approximations: Physical GKP states require finite-energy approximations because ideal states are nonnormalizable and have infinite energy, although finite-energy noise need not prevent fault-tolerant use.A common approximation replaces delta peaks with finite-width Gaussians and applies a Gaussian envelope.
  • State preparation and cluster states: GBS prepares non-Gaussian states, including GKP qubits, probabilistically from Gaussian resources and photon-number-resolving detection, while shared CV controlled-Z gates enable hybrid cluster states.The same CV CZ gate appears in both GKP encoding and canonical CV cluster-state generation.
  • Cluster states and fault tolerance: Surface-code-style fault tolerance can be implemented optically through foliated cluster states; the RHG lattice combines layered primal and dual sheets with nearest-neighbor structure and computational thresholds near 1%.The RHG lattice is universal for measurement-based quantum computation and supports fault-tolerant computation through entangled layers.

3 An Architecture for Photonic Quantum Computing With Hybrid Resource States

The architecture combines probabilistically generated GKP qubits with momentum-squeezed states in a hybrid resource state, generated and entangled across one temporal and two spatial dimensions. Multiplexing boosts GKP availability, while integrated optical modules create the cluster state for measurement-based computation.

  • Architecture: Four modules prepare single-mode states, multiplex probabilistic GKP sources, generate deterministic entangling operations, and perform homodyne-based measurement.The first two modules prepare single-mode states, while entanglement and measurement are assigned to later modules.
  • Hybrid resource states: The architecture encodes information in a hybrid resource state containing GKP qubits and momentum-squeezed vacuum modes.When GKP generation fails, a squeezed state substitutes for the missing qubit while preserving the entanglement structure.
  • Multiplexed state generation: Multiplexed GBS devices raise the probability of preparing a GKP state by running multiple probabilistic sources in parallel.A binary tree of 2×2 switches routes a successful GKP state to the output; if all sources fail, a momentum-squeezed state is swapped in.
  • Temporal cluster generation: The 1D temporal cluster is generated by sending successive GKP or momentum-squeezed pulses through a delay loop with CZ interactions between neighboring modes.The delay line is set to one clock period, and integrated implementations are preferred because it must remain phase stable.
  • RHG lattice generation: Additional spatial CZ gates connect an array of temporal cluster sources to generate the three-dimensional RHG lattice.Alternating source and gate sets operate on even and odd clock cycles to build the lattice layers and their connections.
  • Photonic QPU: The photonic QPU performs measurement-based computation using homodyne detector cells and fast classical control.The generated hybrid resource state is measured to implement the computation.

4 Error Correction for a Quantum Memory

The quantum-memory error-correction procedure prepares a noisy hybrid RHG resource, homodynes its modes, and applies inner continuous-variable and outer qubit decoding. Its noise model captures correlations introduced by CZ gates and squeezed-state substitutions, which the inner decoder can exploit.

  • Error-correction procedure: Quantum error correction initializes each RHG node as either a noisy GKP state with probability 1 −p0 or a finitely squeezed momentum eigenstate with probability p0.Both node types are characterized by the noise variance parameter δ.
  • Error-correction procedure: The procedure measures p-homodyne outcomes, converts real-valued data into binary qubit outcomes with an inner decoder, and applies an RHG outer decoder.The outer decoder returns a recovery operation with a continuous-variable implementation.
  • Error-correction procedure: Continuous-variable feed-forward combined with the qubit recovery operation produces the complete recovery operation, which can be tracked in software.This completes the correction procedure on the active physical modes.
  • Error model: The noise model treats finite-energy GKP states and squeezed-state substitutions as Gaussian noise channels whose propagation through CZ gates creates correlated noise.The correlations depend on the locations of momentum-squeezed states and the lattice-dependent CZ pattern.
  • Error model: Correlated Gaussian noise converts each ideal p-space lattice point into a distribution centered at that point with covariance matrix eΣp.The covariance contains correlations that inform the choice of inner decoder.
  • Inner decoder: For a momentum-squeezed node surrounded by four GKP neighbors, correlation-aware decoding yields either identity or a correlated four-body Z ring that acts trivially on the code space.Naive independent binning instead produces a high-strength dephasing channel on the neighboring GKP qubits.

5 Fault-Tolerant Universal Quantum Computation

The architecture implements fault-tolerant logical computation by encoding qubits in lattice patches and using homodyne measurements, patch surgery, ancillas, and injected magic states. Compilation reduces the computation to multi-qubit Pauli measurements and magic-state-assisted non-Clifford operations.

  • Logical Qubits: Logical qubits are encoded in continuous lattice patches surrounded by q-measured gaps, with patch deformation enabling movement of encoded information.Patch deformation changes q measurements to p in the surrounding gap and requires sufficient error-correction rounds.
  • State Initialization: Single logical states are initialized as |¯+⟩ or |¯0⟩ by measuring first-layer ancillas in the p or q quadrature, respectively.Fault-tolerant initialization uses a number of layers scaling linearly with code distance followed by error correction.
  • Logical Operators and Measurements: Homodyne measurements implement logical Pauli measurements: q and p measurements on primal sheets yield logical ¯Z and ¯X, with the bases swapped on dual sheets.Logical outcomes depend on parity along corresponding chains, and error correction is required for reliable inference.
  • Multi-Qubit Operations: Merging and splitting patches implement entangling operations and multi-qubit Pauli measurements by changing measurements in the gap between patches.Fault-tolerant merging or splitting requires error-correction rounds; splitting maps α|¯0⟩+β|¯1⟩ to α|¯0¯0⟩+β|¯1¯1⟩.
  • Multi-Qubit Operations: An ancillary patch of physical |+⟩ states measures tensor products of Pauli operators by merging with the logical patches and correcting the resulting stabilizer data.The ancilla patch carries no logical information and is coupled through the relevant patch boundaries.
  • Magic-State Injection and Computation: Non-Clifford operations require injected and distilled logical magic states, while compilation commutes Clifford operations to the end and absorbs them into multi-qubit Pauli measurements.Physical magic states are post-selected and enlarged through error correction before use in logical gates.

6 Threshold Estimation for a Quantum Memory

The paper estimates fault-tolerance thresholds by simulating a quantum memory across lattice sizes and noise parameters, applying inner and outer decoding before checking correction success. The simulations provide thresholds for finite squeezing and probabilistic GKP-state replacement.

  • Threshold Procedure: The architecture has a threshold under finite-squeezing errors and a range of swap-out probabilities, established by complete quantum-memory error-correction simulations.The simulations model approximate GKP states and probabilistic sources of GKP qubits.
  • Threshold Procedure: Threshold estimation varies lattice distance d, noise variance δ, and swap-out probability p0, using Monte Carlo trials followed by inner decoding, outer decoding, and success checks.For the RHG lattice, the number of qubits scales as n = O(d3).
  • Threshold Procedure: Correction succeeds when the physical error combined with recovery is a stabilizer and fails when their product is a non-trivial logical operator.This success criterion follows the recovery operation and determines the logical error rate used in threshold estimation.
  • Threshold Results: At p0 = 0.06, incorporating analog homodyne information changes the threshold from ∼15.5 dB to ∼12.2 dB, while the modified inner decoder does not significantly change the thresholds.Below threshold, failure rates using the modified inner decoder are equal to or lower than those from the alternatives.
  • Threshold Results: 10.5 dB is the observed error threshold for the RHG-GKP code with approximate GKP states and no swap-outs.The threshold is reported as ∆dB = −10 log10(δ) ≈10.5 dB using standard binning and matching-graph weights.
  • Threshold Results: At 15 dB squeezing, simulations suggest a swap-out threshold of p0 ≈0.133, with decoder noise-variance tolerance better for p0 ≲0.19.The threshold is a line in (δ, p0) parameter space rather than a single point.

7 Open Problems

The paper identifies open problems in hardware, encoding, noise modeling, and decoding. Key boundaries include incomplete physical noise models, difficult high-quality GKP preparation, and decoders that may still be substantially improved.

  • Hardware: The architecture still requires a passive implementation of its active CZ gates to simplify the computational module and reduce experimental requirements.The current CZ transformations require in-line squeezing and displacements.
  • Hardware: Reliable preparation of GKP qubits with teeth squeezed at or above 15 dB remains an important experimental goal.This motivates improving GBS-based state-generation methods.
  • Noise Models: The main simulations use a Gaussian state-preparation noise model and do not yet incorporate non-Gaussian GBS-state features, CZ-gate noise, homodyne-detector noise, or transmission losses.More realistic yet tractable noise models are identified as requiring further analysis.
  • Encoding: Alternative bosonic encodings and hybrid resource states remain unexplored possibilities for overcoming low GKP-generation probabilities or large multiplexing requirements.Candidate encodings must retain compatibility with swap-outs and the relevant optical entangling operations.
  • Decoding: Current decoding leaves exploitable information unused, including real-valued homodyne data at the outer-decoder level and complicated p-squeezed-state arrangements.Analog and maximum-likelihood decoding may further reduce squeezing thresholds, while improved inner and outer decoders may increase swap-out thresholds.
  • Decoding: Fast decoding of real-valued homodyne signals is needed to exploit architecture clock speeds that could reach GHz.The decoder must process signals quickly enough to change homodyne local-oscillator phases during computation.

8 Summary and Technological Advantages

The architecture combines a hybrid resource state with tailored decoding to support fault-tolerant photonic computation while reducing fabrication, cryogenic, and operating demands. Its modular design separates hardware tasks, and its numerical results identify tolerable swap-out regimes.

  • A hybrid resource state combines bosonic qubits and squeezed states, with a two-tier decoder using continuous- and discrete-variable syndrome data.
  • 23.6% is the approximate upper bound on swap-out probability below which fault-tolerant quantum computation is possible.
  • At 15 dB squeezing, the maximum tolerable swap-out probability is approximately 13.3%.The 15 dB value has been achieved in free-space implementations, while integrated sources have demonstrated 8 dB.
  • Modular task allocation assigns state preparation, multiplexing, cluster generation, and measurement to specialized chips with different hardware requirements.State-generation and cluster-stitching circuits can be non-reconfigurable, while cluster measurements require reconfigurable homodyne components.
  • The architecture needs only limited cryogenic infrastructure: state-generation chips may be cryogenic, while the remaining architecture can operate at room temperature.Small commercially available cryogenic technology can replace custom-built large cryostats.
  • Homodyne-detection timescales can set the clock speed, potentially enabling very low-loss on-chip delay lines.The proposal identifies homodyne detection as faster than photon-number-resolving detectors used in multiplexing.

A Noise Model for a Hybrid RHG Lattice Operating as a Memory

The appendix specifies the error model used to justify the inner decoder and summarizes the numerical procedure for evaluating the hybrid RHG memory.

  • The appendix gives full details of the error model summarized in Section 4.1.
  • The error model is used to justify the choice of inner decoder.
  • The model supports analysis of a hybrid RHG lattice operating as a quantum memory.

A.1 Noisy Initial States

The noisy-initial-state model represents GKP and momentum-squeezed modes with state-dependent additive Gaussian noise, while tracking their distinct covariance structures and assumptions.

  • The cluster contains either |+⟩gkp states or momentum-squeezed states, with state preparation modeled by a single-mode additive Gaussian noise channel.
  • Each mode is assigned a state-dependent covariance matrix, using Ygkp for GKP modes and Yp for momentum-squeezed modes.
  • The initialization samples a momentum-squeezed state with probability p0 and a |+⟩gkp state with probability 1−p0.
  • The noise channel broadens ideal GKP phase-space peaks into Gaussian distributions, with the covariance determined by the populated state.
  • Yp(ϵ, δ) converts |+⟩gkp into a classical mixture of noisy p-squeezed states with p-quadrature variance δ/2.
  • The model treats preparation noise as a classical channel related to pure loss, but assumes noiseless CZ gates and measurements.The authors state that imperfections in those modules would likely reduce the error threshold.

A.2 Propagation of Noise in the Cluster State Preparation

Cluster-state preparation applies CZ gates according to the RHG lattice connectivity and propagates the resulting noise through the corresponding symplectic transformation.

  • Each node is prepared as a momentum-squeezed state with probability p0 or as a |+⟩gkp state with probability 1−p0.
  • Perfect CZ gates are applied to every pair of nodes connected by an RHG-lattice edge, with selected gates inverted for the CV toric-code convention.
  • The adjacency matrix ARL records which optical modes are entangled by CZ gates and defines the symplectic transformation linking the N modes.
  • The full noise matrix evolves under the cluster-preparation symplectic transformation, after which the momentum covariance is isolated for subsequent analysis.

A.3 Probability Distribution in Momentum Space

The hybrid lattice’s momentum-space distribution is modeled as an ideal RHG lattice broadened by correlated Gaussian noise, allowing homodyne outcomes to be decoded into recovery operations.

  • The hybrid lattice is treated as an ideal RHG cluster state followed by a correlated multimode Gaussian noise channel.
  • The momentum-space distribution under Gaussian noise is obtained by convolving the noiseless distribution with the noise channel’s marginal Gaussian distribution.
  • Gaussian noise broadens each ideal lattice point at n√π into a Gaussian function with covariance eΣp.
  • Homodyne momentum outcomes are sampled from the noise matrix eΣp, after which a classical decoder produces the net recovery operation.

B Optical Components for GKP Qubit Operations

GKP Clifford operations are implemented with Gaussian optical components, while the non-Clifford T gate is implemented by teleportation using a GKP magic state.

  • Clifford operations: GKP Clifford gates and measurements correspond to CV Gaussian gates and measurements, although the architecture implements them measurement-based.
  • Non-Clifford operation: The T gate requires non-Gaussian resources and is implemented through gate teleportation with a GKP magic state.
  • Clifford operations: Pauli X and Z gates use displacements by √π along q and p, respectively, while the Hadamard gate uses a π/2 phase-space rotation.
  • Clifford operations: Homodyne measurement settings select the phase-space measurement axis: φ = 0 gives q homodyne and φ = π/2 gives p homodyne.
  • Non-Clifford operation: The feedforward phase gate is applied when the ancillary mode detects |−⟩gkp through a qubit X measurement using CV p homodyne.

C Heuristic Weights for the Outer Decoder

The outer decoder uses heuristic site weights to account for correlated phase flips caused by p-squeezed neighbors, improving on marginal-error weighting.

  • Decoder weighting: Marginal phase-flip probabilities do not capture the correlated phase flips expected when p-squeezed states replace lattice nodes.
  • Weight construction: The heuristic weight choice is motivated by analyzing a central node with zero to four p-squeezed neighbors while surrounding next-layer nodes remain GKP states.
  • Weight construction: The assumptions used to select weights may differ from simulation parameters, making the resulting weights usable but potentially suboptimal.
  • Noise mechanism: Replacing GKP modes with p-squeezed states causes random q-quadrature displacements to propagate onto neighboring p-quadratures through CV CZ gates.
  • Decoder failure modes: 25%, 33%, and 40% approximate readout-violation probabilities occur when two, three, and four neighboring displacements are nonzero, respectively.
  • Decoder failure modes: A single p-squeezed neighbor produces a closed ring of four Z gates, whereas multiple such neighbors can violate the parity condition required for a closed ring.
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