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Time-Domain Multiplexed 2-Dimensional Cluster State: Universal Quantum Computing Platform

Warit Asavanant, Yu Shiozawa, Shota Yokoyama, Baramee Charoensombutamon, Hiroki Emura, Rafael N. Alexander, Shuntaro Takeda, Jun-ichi Yoshikawa, Nicolas C. Menicucci, Hidehiro Yonezawa, Akira Furusawa

arXiv:1903.03918v2quant-ph

TL;DR

One-dimensional cluster states support only one-input MBQC, while two-dimensional states can support universality, making large-scale 2D cluster-state generation essential for scalable universal MBQC. The paper uses time-domain multiplexing to generate and measure continuous-variable optical cluster states, and verifies a resource for universal MBQC capable of about 5,000 operation steps on 5 input modes.

  • Problem

    One-dimensional cluster states support only one-input MBQC, while large-scale two-dimensional cluster-state generation is essential for scalable universal MBQC.

    Source: p. 1

  • Method

    The authors use time-domain multiplexing to continuously generate and immediately measure a two-dimensional cluster state with a continuous-variable optical system.

    Source: p. 1

  • Results

    The experiment generated and verified a two-dimensional cluster state for universal CV MBQC, capable of about 5,000 operation steps on 5 input modes.

    Source: pp. 2, 4

  • Takeaways & Limitations

    The continuously generated and immediately measured state can be arbitrarily long even with finite coherence time of the light source.

    Source: p. 1

    Ancillary-state injection for non-Gaussian operations uses part of the cluster state, reducing the number of input modes it can support.

    Source: S6.4.1. Cubic phase gate with… · p. 21

Abstract

from arXiv · show

Quantum computation promises applications that are thought to be impossible with classical computation. To realize practical quantum computation, the following three properties will be necessary: universality, scalability, and fault-tolerance. Universality is the ability to execute arbitrary multi-input quantum algorithms. Scalability means that computational resources such as logical qubits can be increased without requiring exponential increase in physical resources. Lastly, fault-tolerance is the ability to perform quantum algorithms in presence of imperfections and noise. A promising approach to scalability was demonstrated with the generation of one-million-mode 1-dimensional cluster state, a resource for one-input computation in measurement-based quantum computation (MBQC). The demonstration was based on time-domain multiplexing (TDM) approach using continuous-variable (CV) optical flying qumodes (CV analogue of qubit). Demonstrating universality, however, has been a challenging task for any physical system and approach. Here, we present, for the first time among any physical system, experimental realization of a scalable resource state for universal MBQC: a 2-dimensional cluster state. We also prove the universality and give the methodology for utilizing this state in MBQC. Our state is based on TDM approach that allows unlimited resource generation regardless of the coherence time of the system. As a demonstration of our method, we generate and verify a 2-dimensional cluster state capable of about 5,000 operation steps on 5 inputs.

METHOD · S1. EXPERIMENTAL SETUP

The setup generated a two-dimensional cluster state using four squeezed-vacuum OPOs, an interferometer with two delay lines, and balanced-homodyne measurements. Measurements showed negligible overlap between adjacent temporal modes, supporting their treatment as independent orthogonal quantum modes.

METHOD

  • OPOs with approximately 80 MHz bandwidth provided the squeezed-vacuum sources.Source: METHOD · p. 5

S1. EXPERIMENTAL SETUP

  • A 1.4 W, 860 nm continuous-wave Ti:Sapphire laser supplied the experiment, including about 1 W for generating 430 nm pump light and 60 mW for local oscillators.With 1 W pump power, the second-harmonic generator produced approximately 460 mW of 430 nm light.Source: S1. EXPERIMENTAL SETUP · p. 7
  • Four triangle-shaped OPO cavities used 10 mm PPKTP crystals, with 1100 mm build-up cavities amplifying their pump beams.Each OPO had intracavity losses of about 0.2–0.3%, equivalent to 1.4–2.1% external loss.Source: S1. EXPERIMENTAL SETUP · p. 7
  • State generation used five 50:50 beamsplitters and two delay lines, with a 12 m free-space short delay and a 40 m fiber long delay for five 40 ns wave packets.The fiber transmittivity was about 90%, and beamsplitter interferometric visibility was around 96–99%.Source: S1. EXPERIMENTAL SETUP · pp. 7–8
  • Relative interferometer phases were locked at 13 places by aligning probe-beam ellipses, using synchronized 16-channel RF signals to drive frequency shifts and demodulation.Tilt locking controlled the cavity lengths for the SHG, MCC, four OPOs, and four build-up cavities.Source: S1. EXPERIMENTAL SETUP · p. 8
  • Balanced-homodyne verification used detectors with around 100 MHz bandwidth, approximately 9.5 mW local-oscillator power, and greater than 97% visibility.Fifth-order high-pass filters with a 4 MHz cutoff removed probe-beam signals; 100 MHz low-pass filters provided anti-aliasing.Source: S1. EXPERIMENTAL SETUP · p. 8
  • Data were sampled at 1 GHz over 12,000 frames, using 40 data points per temporal mode and 40 ns wave packets shaped to minimize adjacent-mode correlation.The wave-packet shape was designed to account for the electrical filters and finite homodyne-detector bandwidth; its parameters were optimized to maximize squeezing while maintaining mode independence (Ref. 3).Source: S1. EXPERIMENTAL SETUP · p. 8
  • Adjacent temporal modes had negligible overlap, confirming that each could be treated as an independent, orthogonal quantum mode.The overlap was assessed using correlation coefficients calculated from shot noise measured with each homodyne detector.Source: Figure S2 · p. 8, S1. EXPERIMENTAL SETUP · p. 8

S2. THEORY FOR GENERATION OF TWO-DIMENSIONAL CLUSTER STATE · S2.1. Nullifiers and Graph

The section derives the nullifiers and graph representation of the generated two-dimensional cluster state. It shows how the graph is constructed and how its nullifiers and geometry relate to computational capability and finite squeezing.

S2.1. Nullifiers and Graph

  • Pure Gaussian states are represented by graphs whose nodes correspond to modes and whose graphs correspond one-to-one with covariance matrices, defining the state up to local displacements.The paper’s shorthand omits self-loops and represents edge magnitude by C and its sign by edge color.Source: S2.1. Nullifiers and Graph · p. 9, Figure S3 · p. 9
  • The derivation starts with p-squeezed and x-squeezed vacuum states from the four OPOs and tracks their nullifiers through the first three beamsplitters.In the ideal infinite-squeezing case, OPO-A and OPO-C begin in |p = 0⟩, while OPO-B and OPO-D begin in |x = 0⟩.Source: S2.1. Nullifiers and Graph · p. 9
  • Two delay lines and beamsplitters BS-4 and BS-5 connect square clusters formed across four spatial modes into a two-dimensional cluster state.The square clusters arise after BS-1 through BS-3.Source: S2.1. Nullifiers and Graph · p. 9, Figure S4 · p. 10
  • Local Fourier rotations on half the modes convert the state’s nullifiers to the standard p−∑gᵢxᵢ form without changing its computational capability, only the measurement basis.The graph structure is preserved, although its edge weights change.Source: S2.1. Nullifiers and Graph · pp. 9–10
  • The graph has an infinitely long helical boundary, with N nodes per turn for N inputs, enabling unlimited modes and operations in the ideal limit or with proper error correction.At finite squeezing, the graph also has self-loops whose weights match those of the EPR state.Source: S2.1. Nullifiers and Graph · p. 10, Figure S4 · p. 10
  • For finite squeezing, the state’s nullifier values correspond to the original squeezing from each OPO in the ideal case without optical loss or experimental imperfections.The nullifiers become nonzero at finite squeezing.Source: S2.1. Nullifiers and Graph · pp. 10–11

S2.2. Derivation of inseparability criteria

The section derives a nullifier-variance threshold for verifying inseparability in the 2-D cluster state. It applies the van Loock–Furusawa criterion across the state’s two connectivity directions and the relevant mode bipartitions.

S2.2. Derivation of inseparability criteria

  • Below −4.5 dB relative to shot noise for each nullifier establishes inseparability; the 1-D threshold is −3 dB, with the higher 2-D threshold attributed to greater complexity and dimensionality.The experimental criterion compares each measured nullifier variance with the derived threshold.Source: S2.2. Derivation of inseparability… · p. 11
  • The van Loock–Furusawa criterion establishes inseparability when the nullifier variances fall below a finite squeezing threshold.The section proposes measuring nullifiers and checking their variances against that threshold.Source: S2.2. Derivation of inseparability… · p. 11
  • The 2-D test must check connectivity in two directions: circumference nullifiers and nullifiers along the helix’s infinite length.The 1-D procedure was repeated for each temporal index; the 2-D case requires checking both directions.Source: S2.2. Derivation of inseparability… · p. 11
  • For the circumference-direction nullifiers, the derivation must negate 62 bipartitions, so the paper calculates them systematically and presents one half, with the other derived similarly.The six modes are grouped as one-versus-five, two-versus-four, or three-versus-three.Source: S2.2. Derivation of inseparability… · p. 11

S2.2.1. One modes and five modes · S2.2.2. Two modes and four modes

The squeezing associated with each nullifier depends on the selected mode or mode pair, ranging from 0 dB to 4.5 dB in these cases.

S2.2.1. One modes and five modes

S2.2.2. Two modes and four modes

S2.2.3. Three modes and three modes

The section establishes 4.5 dB of squeezing per nullifier as a sufficient inseparability condition, matching the expected worst-case threshold for this state. Certain three-mode configurations require only 1.2 dB per nullifier.

Three modes and three modes

  • 1.2 dB of squeezing per nullifier is obtained when two modes are (A,k) and (B,k), with the third from the specified adjacent-time or offset-mode set.The third mode is one of (A,k +1), (B,k +1), (C,k +N), or (D,k +N).Source: S2.2.3. Three modes and three modes · p. 12
  • 1.2 dB of squeezing per nullifier is also obtained when all three modes come from the specified adjacent-time or offset-mode set.That set is (A,k +1), (B,k +1), (C,k +N), and (D,k +N).Source: S2.2.3. Three modes and three modes · p. 12
  • For mode combinations outside those two cases, the nullifiers correspond to 4.5 dB of squeezing per nullifier.The section relates this result to the −10log10 g necessary squeezing criterion for Gaussian cluster states whose graph is bipartite and self-inverse.Source: S2.2.3. Three modes and three modes · pp. 12–13
  • 4.5 dB of squeezing per nullifier is sufficient for inseparability and is the expected worst-case necessary threshold for this state.The threshold is sufficient, not proven necessary in general; the section notes that a lower threshold may exist. The expectation follows from the −10log10 g criterion for bipartite, self-inverse Gaussian cluster states and the state's equivalence to one up to local phase shifts.Source: S2.2.3. Three modes and three modes · pp. 12–13

S3. MEASUREMENT OF THE LENGTH OF TWO OPTICAL DELAY LINE

This section describes measuring and tuning both optical delay lines precisely, a requirement for generating the 2D cluster state. Phase-difference measurements yield lengths of 11.85 m and 59.75 m, in a 0.99 : 5 ratio.

S3. MEASUREMENT OF THE LENGTH OF TWO OPTICAL DELAY LINE

S4. EFFECTS OF OPTICAL LOSSES AND LENGTH MISMATCH ON NULLIFIERS · S5. POWER SPECTRAL FROM HOMODYNE DETECTORS

The sections analyze how optical loss and delay-line mismatch affect nullifiers, and show how homodyne power spectra can diagnose phase locking and quadrature measurements in the two-dimensional cluster-state system.

S4. EFFECTS OF OPTICAL LOSSES AND LENGTH MISMATCH ON NULLIFIERS

  • With no length mismatch, detection efficiencies η = 0.75 and 0.80 predict nullifiers of −4.96 dB and −5.62 dB, respectively, in good agreement with experiment.The model assumes equal detection efficiency across paths and a normalized OPO pump amplitude ξ = 0.65.Source: S4. EFFECTS OF OPTICAL LOSSES AND… · p. 15
  • In the ideal delay-match case, optical loss acts like additional loss on the squeezed-light source, while mismatch mixes anti-squeezing into nullifiers and degrades temporal-mode orthogonality and independence.The two-dimensional setup uses two delay lines, with the longer ideally an integer multiple of the shorter; this mismatch issue does not arise in the one-dimensional case.Source: S4. EFFECTS OF OPTICAL LOSSES AND… · pp. 14–15
  • Actual path-dependent detection efficiencies and phase offsets or fluctuations can further degrade nullifiers, although the predictions agree with experiment without modeling these effects or length mismatch.Measured detection efficiencies vary by path from about 0.75 to 0.80; phase locking involves pump beams and beam splitters used for generation and homodyne measurements.Source: S4. EFFECTS OF OPTICAL LOSSES AND… · p. 15

S5. POWER SPECTRAL FROM HOMODYNE DETECTORS

  • Power spectra provide a phase-locking diagnostic that nullifier values cannot: poor nullifiers do not identify the mislocked beam splitter, and measurements in a wrong basis may still appear squeezed.The interferometer contains four light sources and nine beam splitters.Source: S5. POWER SPECTRAL FROM HOMODYNE… · p. 16
  • Oscillatory spectra indicate correct phase locking, with oscillation periods set by delay-line lengths; swapping between x and p spectra confirms the measured quadratures rather than the same quadrature.This quadrature distinction cannot be made from nullifier data alone.Source: S5. POWER SPECTRAL FROM HOMODYNE… · p. 16
  • The spectra arise from frequency-dependent interference in two asymmetric Mach-Zehnder interferometers; including BS-3 is equivalent to 50% loss in one arm, with EPR-pair noise entering instead of vacuum noise.The first two terms in the spectrum describe Mach-Zehnder oscillations, the penultimate term EPR noise, and the last term optical loss.Source: S5. POWER SPECTRAL FROM HOMODYNE… · pp. 16–17
  • Measured homodyne power spectra agree with theoretical predictions, and their shapes support correct phase locking.Figure S8 compares predicted and measured x and p spectra; measurements use 524,288 Fourier points from 1 GS/s recordings averaged 3,000 times.Source: Figure S8 · p. 17, S5. POWER SPECTRAL FROM HOMODYNE… · p. 17

S6. PROOF OF UNIVERSALITY OF TWO-DIMENSIONAL CLUSTER STATE

The section defines the operations needed to establish universality and the control properties desirable for practical MBQC.

S6. PROOF OF UNIVERSALITY OF TWO-DIMENSIONAL CLUSTER STATE

  • The universality proof requires arbitrary one-mode Gaussian operations, a two-mode Gaussian operation, and at least one one-mode non-Gaussian operation; practical MBQC favors easy programming and independent control.The section presents these as requirements to demonstrate and desirable features for actual use.Source: S6. PROOF OF UNIVERSALITY OF… · p. 17

S6.1. Measurement

The section describes how homodyne measurements in the two-dimensional cluster state can be implemented through same-basis measurements after a beam splitter and post-processing.

S6.1. Measurement

  • Same-basis homodyne measurements on modes C and D after a beam splitter are equivalent to measuring without the beam splitter and post-processing the results.This equivalence is illustrated in Fig. S9.Source: S6.1. Measurement · p. 17, Figure S9 · p. 18
  • Post-processing these measurement results also disentangles adjacent modes into separate quantum wires.Source: S6.1. Measurement · p. 18

S6.2. One-mode Gaussian Operations

One-mode Gaussian operations are implemented by isolating a quantum wire from adjacent wires and selecting teleportation measurement bases. The measurement freedoms allow arbitrary one-mode Gaussian operations in two operational steps.

S6.2. One-mode Gaussian Operations

  • Selecting the same teleportation measurement basis in the blue-box circuit disentangles the required quantum wire from adjacent wires, while changing the green-box basis implements the one-mode operation.The resulting operation is equivalent to quantum teleportation plus squeezing; the measurement must be restricted to prevent interactions between adjacent modes.Source: S6.2. One-mode Gaussian Operations · p. 18
  • With finite squeezing and displacements ignored, measuring x in both detectors of the first blue box projects the EPR state to |x = 0⟩; measuring p in the second makes the subsequent circuit a universal squeezer.The passage gives the resulting squeezing parameter as an incomplete expression, so no value is reported here.Source: S6.2. One-mode Gaussian Operations · p. 18
  • Arbitrary one-mode Gaussian operations can be implemented in two operational steps using two degrees of freedom at homodyne-C and homodyne-D.Orthogonal measurement bases at adjacent temporal indices for homodyne-A and homodyne-B produce the additional squeezing gate.Source: S6.2. One-mode Gaussian Operations · p. 18

S6.3. Two-mode Gaussian Operations

Two-mode Gaussian operations are implemented by choosing measurement bases that couple selected adjacent input modes in the cluster state. The method isolates those modes through measurements at k and k + 2, with an example realizing a QND interaction plus squeezing gates.

S6.3. Two-mode Gaussian Operations

S6.4. One-mode Non-Gaussian Operations … S6.5. Feed-forward

The cluster state supports non-Gaussian operations through cubic-phase-state injection or GKP-based magic states. For the described non-Gaussian method, measurement outcomes can account for displacements, leaving adaptive homodyne as the necessary feed-forward operation.

S6.4. One-mode Non-Gaussian Operations

  • Cubic-phase gates outside the cluster complete universality but require an optical switch and homodyne detector, while directly implementing higher-order gates is preferable to combining cubic-phase gates.The paper presents ancillary-state injection and a GKP-based approach as alternatives.Source: S6.4. One-mode Non-Gaussian Operations · p. 20

S6.4.1. Cubic phase gate with ancillary state injection

  • Ancillary cubic-phase-state injection enables non-Gaussian operations for arbitrary error-correction encodings without extra components, but uses cluster modes and reduces supported inputs.The paper says this allocation does not impair computational capability at scale and offers flexibility for higher-order nonlinear measurements.Source: S6.4.1. Cubic phase gate with… · p. 21
  • Replacing one input with a cubic phase state makes the circuit equivalent to the cubic-phase gate of Ref. 12, and existing TDM delays provide adaptive-homodyne time without an additional delay line.The measurement of one mode depends on the prior result for another; the available computation time is determined by the long delay line.Source: S6.4.1. Cubic phase gate with… · p. 21, Figure S12 · p. 21
  • Higher-order nonlinear phase gates can use tunable two-mode QND or beam-splitter operations with suitable ancillary states and homodyne measurements, all realizable with the cluster state.With sufficiently good ancillary states, this approach requires no additional quantum circuitry.Source: S6.4.1. Cubic phase gate with… · p. 21

S6.4.2. Generation of distillable magic state with GKP qubits

  • With GKP encoding, GKP qubits can supply distillable magic states using Gaussian operations and homodyne or heterodyne measurements, without photon counting or other non-Gaussian components.The paper describes GKP qubits, alongside two-dimensional cluster states, as sufficient for both universality and fault tolerance.Source: S6.4.2. Generation of distillable… · p. 21

S6.5. Feed-forward

  • Displacements can be delayed across Gaussian operations but not generally across non-Gaussian operations; for the described ancillary-state method, they can instead be incorporated into adaptive homodyne results.Subtract the displacement value from the homodyne outcome; the circuit illustration shows how past results determine the resulting displacement.Source: S6.5. Feed-forward · p. 22, Figure S13 · p. 22
  • Only adaptive changes to the measurement basis are necessary as feed-forward: corrective displacements can be accounted for in final homodyne results, and this also holds for the current cluster-state setup.The measurement basis uses prior measurement results to account for displacement values.Source: S6.5. Feed-forward · pp. 22–23
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