PaperlayerView exact revision on arXiv

Source-linked AI summary

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

Universal measurement-based quantum computation requires a scalable two-dimensional cluster state, but prior demonstrations were limited to one-input resources. This paper proves and experimentally realizes a time-domain-multiplexed continuous-variable two-dimensional cluster state, generating a resource supporting about 5,000 operation steps on five inputs.

  • Problem

    Demonstrating a resource state that supports arbitrary Gaussian and non-Gaussian operations remains necessary for universal measurement-based quantum computation.

  • Method

    The paper uses time-domain multiplexing with two precisely controlled optical delay lines and proves universality through programmable Gaussian operations, non-Gaussian operations, and adaptive feed-forward.

  • Results

    The generated cluster-state graph has an infinitely long helix, corresponding to an unlimited number of modes, while temporal modes remain orthogonal and independent.

  • Takeaways & Limitations

    The demonstrated two-dimensional cluster state provides a scalable resource architecture for universal measurement-based quantum computation.

  • Takeaways & Limitations

    Ancillary-state injection for non-Gaussian operations consumes part of the cluster state and reduces the number of supported input modes.

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 · COMPETING FINANCIAL INTERESTS · S1. EXPERIMENTAL SETUP

The experiment used squeezed-vacuum OPO sources and a TDM optical setup with nonlinear generation, delay-line interferometry, homodyne verification, and temporal-mode analysis. The authors declare no competing financial interests.

  • METHOD: Optical parametric oscillators with approximately 80 MHz bandwidth served as squeezed-vacuum sources.These sources provided the squeezed vacua used in the optical experiment.
  • COMPETING FINANCIAL INTERESTS: The authors declare no competing financial interests.The supplementary information accompanies the paper “Time-Domain Multiplexed 2-Dimensional Cluster State: Universal Quantum Computing Platform.”
  • S1. EXPERIMENTAL SETUP: A CW 860 nm Ti:Sapphire laser supplied about 1.4 W, including approximately 1 W for SHG and 60 mW for mode cleaning.The SHG generated approximately 460 mW of 430 nm pump light using a KNbO3 nonlinear medium.
  • S1. EXPERIMENTAL SETUP: The setup used four PPKTP-crystal OPOs with 48 mm cavities, 14% output-coupler transmittivity, and 0.2–0.3% intracavity losses.These losses correspond to 1.4–2.1% external loss per OPO.
  • S1. EXPERIMENTAL SETUP: Five 50:50 beamsplitters and two delay lines implemented state-generation interferometry, with a 40 m fiber providing the 200 ns long delay.The short free-space delay was 12 m, and fiber transmittivity was about 90%; beamsplitter visibility was 96–99%.
  • S1. EXPERIMENTAL SETUP: Balanced homodyne detection verified the generated two-dimensional cluster state using a 100 MHz detector, 9.5 mW local oscillator, and >97% interferometric visibility.Probe-beam contributions were electrically removed from the measured signals.
  • S1. EXPERIMENTAL SETUP: Oscilloscope recordings sampled at 1 GHz across 12,000 frames, with 40 data points per temporal mode used to calculate quadrature statistics.The 40 ns wave-packet shape was designed to minimize correlations between adjacent temporal modes.
  • S1. EXPERIMENTAL SETUP: The temporal wave packets had negligible intermode overlap, confirming independent, orthogonal bosonic quantum modes for the measurement.The correlation coefficient was evaluated from shot-noise measurements for each homodyne detector.

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

The section derives nullifiers and the graph structure of the experimentally generated two-dimensional continuous-variable cluster state. It establishes equivalence to the original cluster-state formulation and relates finite-squeezing nullifier noise to the source OPO squeezing.

  • S2.1. Nullifiers and Graph: The generated state is described by a graph whose nodes are modes and whose structure corresponds one-to-one with the Gaussian state's covariance matrix.This graph representation uniquely specifies the pure Gaussian state up to local displacements.
  • S2.1. Nullifiers and Graph: The derivation begins from p = 0 and x = 0 squeezed-vacuum inputs and tracks their nullifiers through the beam-splitter network.The initial assignments are p-squeezed vacua from OPO-A and OPO-C and x-squeezed vacua from OPO-B and OPO-D.
  • S2.1. Nullifiers and Graph: After BS-1 to BS-3, each temporal index forms a square cluster linking the four spatial modes.Two optical delay lines and BS-4 and BS-5 then connect these squares into a two-dimensional cluster state.
  • S2.1. Nullifiers and Graph: Local Fourier transforms on half the modes convert the nullifiers to the standard p−∑g_i x_i form without changing computational capability.The transformation changes the measurement basis and preserves the graph structure while modifying edge weights.
  • S2.1. Nullifiers and Graph: For N = 5, the resulting graph has edge-weight magnitude C = 1/√2 relative to the EPR state.The graph also incorporates the experiment's cylindrical boundary condition and temporal indexing.
  • S2.1. Nullifiers and Graph: In the finite-squeezing case, self-loops appear in addition to inter-node edges, with self-loop weights matching those of the EPR state.Fourier transforms alter edge weights but not the graph's structure.
  • S2.1. Nullifiers and Graph: In the ideal lossless, imperfection-free limit, finite-squeezing nullifier values correspond to the original squeezing supplied by each OPO.The nullifiers are expressed through the squeezing parameters and vacuum quadratures of the source modes.

S2.2. Derivation of inseparability criteria

The section derives an inseparability criterion from cluster-state nullifiers using the van Loock–Furusawa criterion and establishes a −4.5 dB variance threshold relative to shot noise. It also systematizes the required bipartition analysis for the six-mode case.

  • Threshold criterion: −4.5 dB nullifier variance relative to shot noise is sufficient to establish inseparability for this cluster state.The corresponding one-dimensional threshold is −3 dB relative to shot noise; the higher threshold reflects the greater connectivity of the two-dimensional state.
  • Threshold criterion: The van Loock–Furusawa criterion uses cluster-state nullifiers as quadrature combinations and tests inseparability by measuring their variances against a finite squeezing threshold.The criterion establishes inseparability when the experimentally measured nullifier variances fall below the derived threshold.
  • Bipartition analysis: The six-mode analysis must exclude 62 bipartitions, compared with 7 in the one-dimensional case, so the calculation is performed systematically.The derivation considers one-versus-five, two-versus-four, and three-versus-three mode bipartitions, with the complementary nullifier set treated analogously.
  • Normalization: The experimental nullifiers are normalized to vacuum, and the vacuum variance for the relevant X quadrature combination equals four shot-noise units.The same relation applies to P, enabling the threshold to be expressed in decibels relative to shot noise.

S2.2.1. One modes and five modes

For one-mode cases, the nullifiers correspond to either 3 dB or 4.5 dB of squeezing, depending on the mode indices.

  • One modes and five modes: Modes (A,k) or (B,k) correspond to 3 dB of squeezing for each nullifier.
  • One modes and five modes: Modes (A,k +1) or (B,k +1) correspond to 4.5 dB of squeezing for each nullifier.
  • One modes and five modes: Modes (C,k +N) or (D,k +N) correspond to 4.5 dB of squeezing for each nullifier.

S2.2.2. Two modes and four modes · S2.2.3. Three modes and three modes

The nullifier analysis assigns squeezing thresholds to mode configurations and establishes 4.5 dB per nullifier as a sufficient condition for inseparability. For the bipartite, self-inverse Gaussian cluster state, 4.5 dB is also necessary in the worst case.

  • S2.2.2. Two modes and four modes: For two modes from (A,k) and (B,k), each nullifier corresponds to 0 dB of squeezing.This is the first mode configuration analyzed for the two-mode case.
  • S2.2.2. Two modes and four modes: For two modes from (A,k +1), (B,k +1), (C,k +N), and (D,k +N), each nullifier corresponds to 3 dB of squeezing.This configuration defines the second two-mode case.
  • S2.2.2. Two modes and four modes: All remaining two-mode configurations correspond to 4.5 dB of squeezing for each nullifier.These cases are those not included in the first two configurations.
  • S2.2.3. Three modes and three modes: When two modes are (A,k) and (B,k) and the third is from the specified shifted modes, each nullifier corresponds to 1.2 dB of squeezing.The third mode may be one from (A,k +1), (B,k +1), (C,k +N), or (D,k +N).
  • S2.2.3. Three modes and three modes: When all three modes are from the specified shifted modes, each nullifier corresponds to 1.2 dB of squeezing.The shifted modes are (A,k +1), (B,k +1), (C,k +N), and (D,k +N).
  • S2.2.3. Three modes and three modes: All remaining three-mode configurations correspond to 4.5 dB of squeezing for each nullifier.The analysis also uses bipartite and self-inverse graph properties to derive separability conditions between modes.
  • S2.2.3. Three modes and three modes: 4.5 dB of squeezing for each nullifier is a sufficient condition for inseparability, although a lower sufficient threshold may exist.The criterion is motivated by the corresponding necessary threshold for bipartite, self-inverse Gaussian cluster states.
  • S2.2.3. Three modes and three modes: 4.5 dB of squeezing is necessary in the worst case for this state, matching the explicitly demonstrated inseparability threshold.The necessary level is determined by −10log10 g, with g = 1 2 √ for this state.

S3. MEASUREMENT OF THE LENGTH OF TWO OPTICAL DELAY LINE

The two-dimensional cluster-state generator requires two optical delay lines whose lengths have a precise integer relationship. Phase-difference measurements and optical-stage calibration determined the short and long lines as 11.85 m and 59.75 m, with a 0.99:5 ratio.

  • System requirement: The two-dimensional cluster-state system uses two optical delay lines, with the long line designed as an integer multiple of the short line.This precision requirement distinguishes the two-dimensional generator from the one-dimensional case.
  • Measurement method: The delay-line lengths were measured by comparing frequency-dependent phase differences through reference and delay paths, then calibrated using an optical stage.The measurement scheme used an electro-optic modulator and network analyzer.
  • Measurement results: 11.85 m and 59.75 m were measured for the short and long optical delay lines, respectively, yielding a 0.99:5 length ratio.Lengths were obtained from phase-frequency dependence and calibrated with an optical stage.

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

The analysis shows how loss and delay mismatch degrade two-dimensional cluster-state nullifiers, while homodyne power spectra diagnose phase locking, quadrature identification, and agreement with theory. Frequency-domain features also reveal how the interferometer’s delay lines and EPR noise shape the measured state.

  • S4. EFFECTS OF OPTICAL LOSSES AND LENGTH MISMATCH ON NULLIFIERS: The nullifier model assumes the wave-packet width equals the short-delay time and uses a common detection efficiency η for all paths.The long-delay mismatch is defined as ∆τ2 = τ2 − N∆t, and the source spectra depend on squeezing, anti-squeezing, pump amplitude, bandwidth, and intracavity loss.
  • S4. EFFECTS OF OPTICAL LOSSES AND LENGTH MISMATCH ON NULLIFIERS: With no long-delay mismatch, the loss terms simplify to an effective additional source loss of 1−η.This idealized limit provides the baseline for assessing how mismatch introduces further degradation.
  • S4. EFFECTS OF OPTICAL LOSSES AND LENGTH MISMATCH ON NULLIFIERS: Additional delay mismatch mixes anti-squeezing into the nullifiers and degrades temporal-mode orthogonality and independence.The derivation assumes equal detection efficiency across paths, although actual efficiencies vary by path and beam.
  • S5. POWER SPECTRAL FROM HOMODYNE DETECTORS: Frequency-domain analysis provides theoretical and experimental insight into the temporally encoded state despite the interferometer’s four sources and nine beam splitters.This complexity motivates examining detector power spectra in addition to nullifier data.
  • S5. POWER SPECTRAL FROM HOMODYNE DETECTORS: Oscillatory homodyne power spectra verify correct phase locking, with oscillation periods determined by the respective delay-line lengths.The spectra swap between x̂ and p̂, confirming that distinct quadratures—not the same quadrature—are being observed.
  • S5. POWER SPECTRAL FROM HOMODYNE DETECTORS: The complex interferometer’s frequency response reflects delay-line phase shifts, while its 50% effective-arm-loss behavior introduces EPR-pair noise rather than ordinary vacuum contamination.The EPR noise arises from one party of the EPR pairs entering the state, alongside oscillatory Mach–Zehnder contributions.
  • S5. POWER SPECTRAL FROM HOMODYNE DETECTORS: Theoretical and experimental power spectra agree for the two-dimensional cluster state across the homodyne detectors.Fig. S8 compares theoretical x̂ and p̂ spectra with experimental measurements; the theory assumes η = 0.75 and ξ = 0.65.

S6. PROOF OF UNIVERSALITY OF TWO-DIMENSIONAL CLUSTER STATE

The section proves that the generated two-dimensional cluster state is universal for MBQC by establishing its ability to implement the required Gaussian and non-Gaussian operations. It also addresses practical use by requiring independently controllable and easily programmable operations.

  • S6. PROOF OF UNIVERSALITY OF TWO-DIMENSIONAL CLUSTER STATE: Universality requires implementing arbitrary one-mode Gaussian operations, a two-mode Gaussian operation, and at least one one-mode non-Gaussian operation.
  • S6. PROOF OF UNIVERSALITY OF TWO-DIMENSIONAL CLUSTER STATE: The state’s practical MBQC use should support easy operation programming and independent control of every operation.
  • S6. PROOF OF UNIVERSALITY OF TWO-DIMENSIONAL CLUSTER STATE: The section proves universality and discusses how the generated state is used in measurement-based quantum computation.

S6.1. Measurement

Homodyne measurements after a beam splitter are equivalent to measuring without the beam splitter and post-processing the results. In the two-dimensional cluster state, this enables measurements using identical bases and helps disentangle adjacent modes into separate quantum wires.

  • Measurement: Using the same measurement basis for homodyne-C and homodyne-D implements this measurement procedure in the two-dimensional cluster state.
  • Measurement: Homodyne measurements after a beam splitter are equivalent to direct measurement followed by post-processing.This equivalence is illustrated in Fig. S9.
  • Measurement: Post-processing the measurement results also disentangles adjacent modes into separate quantum wires.

S6.2. One-mode Gaussian Operations

The 2-D cluster implements one-mode operations by isolating a quantum wire from adjacent wires through coordinated measurements. With two operational steps, the protocol realizes arbitrary one-mode Gaussian operations, with finite-squeezing noise treated as in Ref.9.

  • One-mode Gaussian Operations: The target quantum wire is disentangled from adjacent wires by selecting the same measurement basis for the relevant temporal modes, preventing unwanted interactions.This isolation is required before applying the one-mode circuit shown in Figure S10.
  • One-mode Gaussian Operations: The resulting one-mode operation is equivalent to quantum teleportation followed by squeezing, subject to measurement-basis constraints that prevent adjacent-mode interactions.For the universal squeezer, adjacent homodyne-A and -B measurements must use orthogonal bases, producing the additional squeezing gate with a ± choice.
  • One-mode Gaussian Operations: Two operational steps suffice to implement arbitrary one-mode Gaussian operations using the homodyne measurement degrees of freedom.Orthogonal measurement bases at adjacent temporal indices determine the additional squeezing sign, while the two degrees of freedom at homodyne-C and -D provide arbitrary Gaussian control.
  • One-mode Gaussian Operations: Finite-squeezing noise can be incorporated using the same treatment as Ref.9.The protocol’s noise analysis therefore follows the cited reference’s framework.

S6.3. Two-mode Gaussian Operations

The cluster state implements two-mode Gaussian operations by selecting measurement bases that entangle chosen adjacent input modes. The resulting operation realizes a QND interaction with an additional squeezing gate on each mode.

  • S6.3. Two-mode Gaussian Operations: Measurement-basis selection enables two-mode operations between chosen adjacent input modes by isolating the corresponding cluster-state subregion.The implementation is represented by an equivalent circuit, including the one-mode operation component.
  • S6.3. Two-mode Gaussian Operations: The resulting operation corresponds to the QND interaction C_x(g) = exp(−ig p_1⊗x_2) with an additional squeezing gate on each mode.
  • S6.3. Two-mode Gaussian Operations: Figure S11 illustrates the basis choice, equivalent circuit, simplified circuit, and an example implementation of the quantum non-demolition interaction.

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

The cluster state supports universal computation through cubic-phase non-Gaussian operations, including ancillary-state injection, while feed-forward can be reduced to adaptive measurement-basis changes and final-result corrections.

  • S6.4. One-mode Non-Gaussian Operations: Cubic-phase gates implemented with additional circuitry outside the cluster state complete the universality proof for this cluster-state architecture.This method follows the approach suggested for bilayer square-lattice cluster states.
  • S6.4.1. Cubic phase gate with ancillary state injection: Ancillary cubic-phase-state injection implements non-Gaussian operations for arbitrary encodings without additional components.The approach offers greater flexibility than preparing separate circuitry for each non-Gaussian operation.
  • S6.4.1. Cubic phase gate with ancillary state injection: Replacing one input mode with a cubic phase state turns the two-mode circuit into a cubic-phase gate when measurement bases are chosen appropriately.The procedure requires adaptive homodyne measurement, with one measurement depending on a prior result.
  • S6.4.2. Generation of distillable magic state with GKP qubits: GKP qubits can serve as distillable magic states using only GKP qubits, Gaussian operations, and homodyne or heterodyne measurements.This avoids photon counting and other non-Gaussian resources for realizing both universality and fault-tolerance.
  • S6.5. Feed-forward: Displacements from prior operations can be delayed through beam splitters and absorbed into homodyne measurements, avoiding advance physical feed-forward.The resulting displacement depends on past measurement results.
  • S6.5. Feed-forward: In standard MBQC, corrective displacements follow measurements, but final homodyne detection can instead account for the accumulated displacement in its results.Thus, corrective displacement operations need not be physically implemented for multimode output measurements.
  • S6.5. Feed-forward: Adaptive measurement-basis changes are the only feed-forward operation necessary when displacement corrections are absorbed into final measurement results.The authors emphasize that this property also holds for the cluster state generated in the current setup.
Loading 1903.03918v2…