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Large-scale silicon quantum photonics implementing arbitrary two-qubit processing

Xiaogang Qiang, Xiaoqi Zhou, Jianwei Wang, Callum M. Wilkes, Thomas Loke, Sean O'Gara, Laurent Kling, Graham D. Marshall, Raffaele Santagati, Timothy C. Ralph, Jingbo B. Wang, Jeremy L. O'Brien, Mark G. Thompson, Jonathan C. F. Matthews

arXiv:1809.09791v1quant-ph

TL;DR

Photonic demonstrations had reached at most two sequential two-qubit entangling operations, short of the three required for arbitrary two-qubit processing. This paper uses silicon photonics and a linear-combination scheme to implement a programmable two-qubit processor, demonstrating process tomography on 98 gates and a two-qubit optimization algorithm.

  • Problem

    Photonic demonstrations had reached at most two sequential two-qubit entangling operations, while arbitrary two-qubit processing requires three, a level of complexity described as impractical to construct and maintain.

    Source: p. 1

  • Method

    The chip implements a linear combination of four tensor products of single-qubit unitaries, using high-dimensional entanglement to realize the equivalent of three sequential entangling gates with two photons.

    Source: p. 2

  • Results

    The chip demonstrated process tomography on 98 two-qubit quantum logic gates and realized a two-qubit quantum approximate optimization algorithm.

    Source: p. 2

  • Takeaways & Limitations

    The demonstrations show the linear-combination scheme is valuable for QIP demonstrations with current state-of-the-art photonics, and that silicon photonics can meet its requirements.

    Source: §3 · p. 6

    The optical linear-combination protocol requires beamsplitters and phase shifters to grow exponentially with qubit count, making the scheme ultimately unscalable.

    Source: §3 · p. 6

Abstract

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Integrated optics is an engineering solution proposed for exquisite control of photonic quantum information. Here we use silicon photonics and the linear combination of quantum operators scheme to realise a fully programmable two-qubit quantum processor. The device is fabricated with readily available CMOS based processing and comprises four nonlinear photon-sources, four filters, eighty-two beam splitters and fifty-eight individually addressable phase shifters. To demonstrate performance, we programmed the device to implement ninety-eight various two-qubit unitary operations (with average quantum process fidelity of 93.2$\pm$4.5%), a two-qubit quantum approximate optimization algorithm and efficient simulation of Szegedy directed quantum walks. This fosters further use of the linear combination architecture with silicon photonics for future photonic quantum processors.

3. Implementing a two-qubit Quantum Approximate Optimization Algorithm for Constraint Sat-

The study demonstrates a p = 1 QAOA on three two-bit constraint-satisfaction problems, using a grid search over angles to identify high-probability solution strings.

3. Implementing a two-qubit Quantum Approximate Optimization Algorithm for Constraint Satisfaction Problems

  • For a general CSP, QAOA seeks a binary string z that approximately maximizes the number of satisfied constraints.A CSP is specified by n bits and m constraints, each involving a small subset of the bits.Source: §3 · p. 4
  • QAOA searches γ and β to maximize the expected constraint score, and measuring the resulting computational-basis state encodes a candidate solution.For p = 1, the experiment searched a fine grid over [0, 2π] × [0, π]; the expected score can be evaluated on a quantum computer and used by an enclosing classical optimization algorithm.Source: §3 · pp. 4–5
  • The experiment used p = 1 QAOA on three two-bit CSPs, including Max2Xor with one constraint, whose target strings were “00” and “11.”For Max2Xor, C(z) = 1/2 + 1/2 z1z2, with z1, z2 ∈ {±1}.Source: §3 · pp. 4–5
  • 99.88±0.10%, 96.98±0.56%, and 99.48±0.27% classical fidelities between experiment and theory were measured for CSP1, CSP2, and CSP3, respectively.For CSP1, “00” or “11” had highest probability; the reported solutions were {1, 1} for CSP2 and {1, 1}, {1, −1}, or {−1, 1} for CSP3.Source: §3 · p. 5

Supplementary Information … 2. Probabilistic circuit for linear-combination of quantum gates

The supplement derives a four-term linear-combination representation of arbitrary two-qubit SU(4) operations and describes a probabilistic circuit that implements a general linear combination of gates.

1. Linear-combination decomposition of SU(4)

  • KAK decomposition expresses an arbitrary two-qubit U ∈ SU(4) using single-qubit gates and a central gate UD.The decomposition identifies P1, P2, Q1, and Q2 as single-qubit gates.Source: §1 · p. 9
  • U is rewritten as four product terms, with coefficients α0–α3 obtained from k1, k2, and k3 and satisfying Σ|αi|² = 1.The terms are αi Ai ⊗ Bi for i = 0, 1, 2, 3.Source: §1 · p. 9
  • Each Ai and Bi is a single-qubit operation formed from P1 or P2, a Pauli operator or identity, and Q1 or Q2.The Pauli choices are σx, σy, and σz for the three non-identity terms.Source: §1 · p. 9

2. Probabilistic circuit for linear-combination of quantum gates

  • The circuit’s unitary ULC implements a general linear combination of gates, with its first row determining the coefficients αi.The displayed matrix and state-evolution expressions are incomplete in the supplied passages.Source: §2 · pp. 9–10
  • Measuring all auxiliary qubits as 0 in the computational basis implements the linear-combination operation ΣαiVi.This is the stated successful measurement condition for the probabilistic circuit.Source: §2 · p. 10
  • The circuit’s success probability is 1/k, with k = 2n.The supplied passage prints the relation as “k = 2n.”Source: §2 · p. 10

3. Deterministic linear-combination circuit for universal two-qubit unitary gates

The circuit implements arbitrary two-qubit unitaries by a deterministic linear combination of four tensor products of single-qubit gates. Measurement-conditioned Pauli corrections recover the same linear combination across control-qubit outcomes.

3. Deterministic linear-combination circuit for universal two-qubit unitary gates

  • The circuit deterministically implements arbitrary U ∈ SU(4) as a linear combination of four tensor products of single-qubit gates.The construction establishes that the required ULC is unitary for arbitrary U ∈ SU(4).Source: §3 · pp. 10–11
  • Computational-basis measurements of the two control qubits select Pauli corrections that recover the same linear-combination operation on the targets.The four outcomes correspond to I ⊗ I, σx ⊗ σx, σy ⊗ σy, and σz ⊗ σz corrections.Source: §3 · p. 11
  • The two control qubits can be replaced by one ququard, and the same approach can implement deterministic linear combinations of four larger gates.With this replacement, ULC is implemented as a single-ququard operation.Source: §3 · p. 11

Appendix B: Optical Linear-combination implementation for universal two-qubit unitary gates · 1. Further discussion of optical linear-combination scheme · 2. Advanced photonic linear-combination circuit design for higher success probability

Appendix B describes the optical linear-combination scheme’s encoding, compares its resource requirements with other approaches, and outlines a circuit design that could improve its success probability under stated conditions.

1. Further discussion of optical linear-combination scheme

  • The optical scheme uses Hilbert-space extension and encodes each photon’s information only in its spatial degree of freedom on the chip.Polarization, time-bin, and orbital angular momentum are also possible encodings in the optical scheme.Source: §1 · p. 11
  • A universal two-qubit circuit requires at least three CNOT gates and eight single-qubit gates, while linear-optical entangling gates without optical nonlinearity are probabilistic.The circuit is shown in Fig. S1(A).Source: §1 · p. 11, Figure S1 · p. 12
  • Pittman’s four-photon CNOT succeeds with probability 1/4, while two-photon protocols succeed with probability 1/9 but require post-selection and cannot be cascaded.Combining two Pittman CNOTs with one post-selected CNOT gives a six-photon circuit with overall success probability 1/144.Source: §1 · p. 12
  • The MBQC alternative requires at least a six-photon cluster state and has success probability 1/128, assuming Type-I fusion succeeds with probability 1/2 each.The required cluster state is shown in Fig. S1(B).Source: §1 · p. 12, Figure S1 · p. 12

2. Advanced photonic linear-combination circuit design for higher success probability

  • A two-qubit unitary succeeds with probability 1/64 on the current chip, potentially rising to 1/4 with certain signal–idler separation and an advanced circuit design.The proposed design uses optical ports left unused by the current chip.Source: §2 · p. 12
  • The advanced SU(4) design uses unused optical ports and succeeds when coincidence detection finds both photons at one of four specified port pairs.The specified pairs are 1, 2; 1′, 2′; 1′′, 2′′; or 1′′′, 2′′′.Source: Figure S2 · p. 12, §2 · p. 12

Appendix C: Details of Device and Experimental Setup … Appendix D: Characterization of the device

The appendices describe a silicon photonic device fabricated on an SOI substrate and the optical setup used to prepare, post-select, and detect photon pairs. At 300 mW pump power, the experiments yielded around 100 two-photon coincidences per second.

1. Device

  • Fabricated with 248 nm lithography on SOI, the device uses 500 nm-wide waveguides and has an effective footprint of approximately 7.1 × 1.9 mm2.The SOI substrate has a 2 µm buried oxide and a 220 nm crystalline silicon top layer; 180 µm × 2.5 µm TiN resistive heaters form the thermo-optic phase shifters.Source: §1 · p. 13
  • The device has 3 dB/cm waveguide propagation loss and around 0.3 dB MMI insertion loss; each phase shifter requires around 57 mW for a 2π shift.The phase shifters have about 800 Ω resistance, require less than 10 mA for a 2π shift, and have ignorable loss compared with the MMIs.Source: §1 · p. 13

2. Setup

  • A 1550.8 nm tunable-laser pump was amplified to 300 mW, filtered to suppress ASE noise, and launched as TE-polarized light through a fiber array.An in-line polarization controller and on-chip TE grating couplers were used for coupling into the chip.Source: §2 · p. 13
  • Two off-chip DWDM filters selected signal and idler photons at 1544.2 nm and 1557.4 nm, respectively, for post-selected measurements.The selected channels were each 6.6 nm from the pump; reversing which filter collected signal and idler could theoretically double coincidence counts but required two additional detectors.Source: §2 · p. 13
  • The signal and idler DWDM filters had full-passband insertion losses of 1.89 dB and 2.88 dB, respectively, and at least 45 dB isolation for non-adjacent channels.Together with on-chip pump filters, the DWDM filters effectively isolated pump light from photon detections.Source: §2 · p. 13
  • Device coupling loss, including input and output, was 13 dB; a Peltier cell with PID control stabilized temperature against heater-induced coupling instability.The coupling loss was estimated by injecting pump light into a long straight waveguide surrounding the device.Source: §2 · p. 13
  • At 300 mW pump power, the experiments obtained around 100 coincidence counts per second, with photon detection efficiency maximized up to 50%.Photons were detected using two fiber-coupled superconducting nanowire single-photon detectors.Source: §2 · p. 13
  • Two-photon coincidences were recorded with a PicoHarp 300 time-interval analyzer using a 450 ps integration window; single-photon dark counts were 800–1500 per second.The integration window was chosen according to the SNSPD jitter time, and the dark counts were described as trivial for two-photon coincidence events.Source: §2 · p. 13

1. Electrical characteristics of the phase shifters

The phase-shifter characterization calibrates applied electrical inputs to optical phase shifts across 62 thermally controlled channels. It measures device resistance and phase-response behavior, and distinguishes independent from cascaded phase shifters.

1. Electrical characteristics of the phase shifters

  • All 62 thermal-optic phase shifters are controlled by current-output equipment, enabling each channel’s current to be set by serial communication from a classical computer.Calibration determines the phase corresponding to each applied voltage or current.Source: §1 · pp. 13–14
  • The fitted resistances are around 800 Ω for most phase shifters and around 580 Ω for the four phase shifters used in the pump filters.An example phase shifter’s measured I-V curve is reported as perfectly linear (Fig. S3).Source: §1 · p. 14, Figure S3 · p. 14
  • Thermal-optic phase shifters have a linear phase–electrical-power relationship but a nonlinear phase–current relationship.The phase-current response is parameterized by the applied current I, resulting phase shift θ(I), and device-specific real parameters φ1 and φ0.Source: §1 · p. 14
  • The 62 phase shifters are classified as independent or cascaded; independent units occupy directly accessible MZIs and assume near-50:50 MMI splitting.An independent MZI in the device achieves an extinction ratio up to 30 dB.Source: §1 · p. 14

2. Calibrating independent phase shifters · 3. Calibrating cascaded phase shifters

Independent phase shifters were calibrated by fitting classical interference fringes, while five-shifter cascaded arrays required multidimensional scans and multiparameter fitting. The fitted cascade parameters describe each array’s response and make its phase shifters reconfigurable.

2. Calibrating independent phase shifters

  • Independent phase shifters were measured by injecting 1550 nm, 20 mW bright light into an MZI input and recording output intensity as heater current rose from 0 to 9 mA in 0.05 mA steps.Source: §2 · p. 14, Figure S4 · p. 15
  • A nonlinear fit of an independent phase shifter’s fringe to Equation (S3) yielded φ1 = 0.1123 and φ0 = 0.3814.Source: §2 · p. 14
  • The device’s four filters, source-intensity settings, and measurement stage use directly calibratable independent phase shifters; source-to-source and linear-combination phases can be calibrated independently after cascade calibration.Source: §2 · p. 14

3. Calibrating cascaded phase shifters

  • Each of the device’s eight cascaded arrays contains five phase shifters for single-qubit-state preparation and gate implementation, but only the whole array’s input and output ports are accessible.Source: §3 · p. 15, Figure S5 · p. 15
  • Calibration scanned the same current range across all five heaters, recorded output power for every configuration, and fitted the resulting 5D fringe; this worked for the device but requires exponentially more data as array size grows.Source: §3 · p. 15
  • For each five-heater current configuration, classical input and output intensities were compared with theoretical values using least-squares minimization to estimate the phase-shifter response parameters.Source: §3 · pp. 15–16
  • Fitting each five-shifter array yielded 16 response parameters that describe its behavior across current configurations, making every phase shifter in the array fully reconfigurable.Source: §3 · p. 16

4. Calibrating on-chip filters

An imbalanced on-chip Mach–Zehnder interferometer filter removes pump light while transmitting the signal and idler wavelengths. Its extinction ratio reaches ∼28 dB, and residual pump light contributes negligible coincidence counts in the functional area.

4. Calibrating on-chip filters

  • The on-chip pump filter uses an imbalanced Mach–Zehnder interferometer to remove redundant pump light from the photon sources.The phase-shifter arm is 54 µm longer than the other arm.Source: §4 · p. 16, Figure S6 · p. 16
  • ∼28 dB extinction separates 1550.8 nm pump light from the 1544.2 nm signal and 1557.4 nm idler wavelengths.The filter has low transmission around the pump wavelength and high transmission around the signal and idler wavelengths.Source: §4 · p. 16
  • With up to 300 mW (24.77 dBm) pump per source, ∼0.475 mW (-3.223 dBm) passes the filter, but remnant-pump pairs cannot generate detected coincidences.Residual pump light generates only trivial SFWM in the functional area; such pairs travel through the same device half, unlike same-source signal and idler pairs.Source: §4 · p. 16

Appendix E: Further details of experimental quantum processing results · 1. Implementing two-qubit quantum logic gates · 2. Quantum process tomography

The appendix details implementations of specific two-qubit gates and reports process fidelities, then describes the quantum process tomography used to characterize the implemented operations.

1. Implementing two-qubit quantum logic gates

  • The gate decompositions use linear combinations of tensor products of Pauli and identity operators, with local single-qubit operations programmed after preparing a maximally path-entangled state.The appendix lists decompositions for CNOT, CZ, and CH.Source: §1 · pp. 16–18
  • SWAP and iSWAP used balanced pumps with relative phases 0 and π/2, respectively, whereas √SWAP required unbalanced pumps.The paper states that this was the first implementation of SWAP and iSWAP.Source: §1 · pp. 17–18
  • Process fidelities were 98.85±0.06% (CNOT), 96.90±0.17% (CZ), 97.57±0.07% (CH), 95.33±0.24% (SWAP), 94.45±0.27% (iSWAP), and 92.41±0.33% (√SWAP).Fig. S8 shows reconstructed process matrices for example gates, overlaid with theoretical predictions.Source: §1 · pp. 16–18, Figure S8 · p. 19
  • The processor implemented 98 two-qubit gates with an average process fidelity of 93.15±4.53%.Each gate’s full process tomography took more than 100 minutes, with a typical integration time of 10 seconds per count.Source: §1 · p. 18

2. Quantum process tomography

  • Quantum process tomography used 16 separable, linearly independent inputs and 16 basis projections per output to reconstruct output density matrices.The input states were formed from combinations of |0⟩, |1⟩, |+⟩, and |+i⟩.Source: §2 · p. 18
  • Maximum-likelihood reconstruction yielded a process matrix from 256 coincidence measurements per gate; fidelity was Tr(χidealχexp), with error bars estimated by Monte Carlo assuming Poissonian photon statistics.Source: §2 · p. 18

3. Demonstrating QAOA

The section details QAOA outcomes for three example constraint-satisfaction problems (CSPs) and reports the circuits and measurement costs used in the experiment.

CSP1

  • For CSP1, the strings {1, 1} and {−1, −1} satisfy its only constraint and each makes C(z)=1.Source: §3 · p. 20

CSP2

  • For CSP2, C({1, 1})=3, the highest value among the four strings, and p=1 QAOA outputs {1, 1} with highest probability.The other strings have C values 1, 1, and -1, respectively.Source: §3 · p. 20

CSP3

  • For CSP3, three strings maximize C at 1, and with proper γ and β, p=1 QAOA outputs each with approximate probability 1/3.The maximizing strings are {1, 1}, {1, −1}, and {−1, 1}.Source: §3 · p. 20

Implementation

  • The three example CSP circuits used one two-qubit entangling gate, half the chip’s capability; each measurement took 15 seconds, and evaluating all 600 parameter pairs took more than 10 hours.Source: §3 · p. 20

4. Simulating Szegedy quantum walk

The section analyzes when Szegedy quantum walks (SQWs) are periodic and reports experimental probability distributions for walks with varied weights and initial states. Selected symmetric weights show quasi-periodic behavior.

4. Simulating Szegedy quantum walk

  • An SQW is periodic exactly when all eigenvalues of its single-step operator are roots of unity with a common period; its period is their periods’ lowest common multiple.For the analyzed two-node complete graph, the reported periodic cases have periods of 6, 4, 6, and 2 steps, and two further cases have periods of 4 and 6 steps; the weight expressions are unclear in the supplied text.Source: §4 · p. 21
  • Experiments implemented the first 200 steps for varied weights and three initial states, measuring node-1 probabilities and calculating average distribution fidelities.The average fidelities are listed in Table I, but their values are not provided in the supplied passages.Source: §4 · p. 21, Table I · p. 21
  • Symmetric-weight SQWs with α = β = 0.43, 0.45, and 0.47 showed quasi-periodic probability distributions in the first 100 steps.The experiments measured the probability of the walker being at node 1; the reported behavior may be useful for designing SQW-based quantum algorithms.Source: §4 · p. 21, Figure S9 · p. 22

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