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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
TL;DR
Implementing quantum algorithms such as QAOA and Szegedy quantum walks requires suitable quantum-computer circuits. This work uses a CMOS-fabricated silicon-photonic device with reconfigurable linear optics to demonstrate both algorithms and achieve high-fidelity walk simulations.
Problem
Quantum approximate optimization and Szegedy quantum-walk algorithms require quantum-computer implementations, including efficient circuits for the walk itself.
Method
The processor uses a linear-combination scheme with reconfigurable silicon photonics to optimize QAOA angles and construct repeated circuits for Szegedy quantum walks.
Results
The device experimentally produced solutions for three 2-bit CSPs and simulated Szegedy walks with state fidelity 93.95±2.52% against theoretical predictions.
Takeaways & Limitations
A single CMOS-fabricated silicon-photonic device supports QAOA demonstrations and efficient simulations of directed, weighted-graph quantum walks.
Takeaways & Limitations
The optical linear-combination protocol requires exponentially increasing numbers of beam splitters and phase shifters as the qubit count grows.
Abstract
from arXiv · showhide
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 processor applies p = 1 QAOA to three two-bit constraint satisfaction problems and experimentally simulates Szegedy quantum walks on weighted two-node graphs. These demonstrations achieve high agreement with theoretical predictions while exposing the optical scheme’s scalability boundary.
- QAOA for constraint satisfaction problems: QAOA searches two angles, γ and β, over [0, 2π] × [0, π] to maximize ⟨γ, β|C|γ, β⟩ and identify satisfying bit strings.For p = 1, the optimized computational-basis state encodes solutions to the CSPs.
- QAOA for constraint satisfaction problems: Three two-bit CSPs were experimentally implemented with p = 1, including the 2-bit Max2Xor problem.The corresponding QAOA circuits are shown for CSP1, CSP2, and CSP3.
- Szegedy quantum walks: 98.46±0.04%, 98.48±0.04%, 98.02±0.04%, and 98.35±0.15% were the reported average fidelities for four SQW weight settings.The settings included symmetric, non-perfectly periodic, and asymmetric cases.
- QAOA for constraint satisfaction problems: 99.88±0.10%, 96.98±0.56%, and 99.48±0.27% were the classical fidelities for the three QAOA experiments, respectively.The measured solution distributions agreed closely with theoretical results for CSP1, CSP2, and CSP3.
- Szegedy quantum walks: Szegedy quantum walks were implemented on a weighted two-node directed graph using a circuit for the single-step walk operator.Repeating the circuit provides an implementation of multiple-step walks.
- Discussion: The linear-combination optical network requires beamsplitters and phase shifters to increase exponentially with qubit number, making the scheme ultimately unscalable.The authors nevertheless describe it as suitable for near- and mid-term situations where photonic components are easier to create than qubits.
Supplementary Information
The supplementary material derives a linear-combination representation of arbitrary SU(4) operations and describes the probabilistic circuit that implements it. Four coefficient-weighted tensor products of single-qubit operations are combined when auxiliary qubits yield the required measurement outcome.
- Linear-combination decomposition: An arbitrary U ∈ SU(4) is decomposed into four coefficient-weighted products A_i ⊗ B_i of single-qubit operations.The coefficients α0, α1, α2, and α3 are obtained from the KAK decomposition parameters.
- Linear-combination decomposition: The operators A_i and B_i are constructed from single-qubit gates P1, P2, Q1, Q2 and Pauli matrices.The listed operations use I, σx, σy, and σz for i = 0, …, 3.
- Probabilistic circuit: The probabilistic circuit implements the linear combination Σ_i α_iV_i when all auxiliary qubits are measured as 0.The success probability is 1/k, where k = 2^n.
- Probabilistic circuit: The first row of U_LC determines the linear coefficients α_i, while the remaining rows are selected to make U_LC unitary.The supplementary construction specifies the matrix entries used by the circuit.
3. Deterministic linear-combination circuit for universal two-qubit unitary gates
The circuit represents an arbitrary U ∈SU(4) as a deterministic linear combination of four tensor products of single-qubit operations. Measurement-dependent Pauli corrections recover the same target operation for every control outcome.
- Circuit construction: The proposed circuit implements a universal two-qubit unitary as a linear combination of four tensor products of single-qubit gates.The coefficients α0, α1, α2 and α3 determine the linear combination, while ULC is constructed as a unitary operation.
- Circuit construction: The two control qubits and target states evolve into four branches associated with I, σx, σy and σz operations.The branches are weighted by the coefficients and correlated with computational-basis control states.
- Correction procedure: Measurement of the control qubits selects the Pauli correction applied to the two target qubits.The four computational-basis outcomes require I⊗I, σx⊗σx, σy⊗σy or σz⊗σz corrections, respectively.
- Extensions: The two control qubits can be replaced by one ququard, and the construction can extend to four larger gates in principle.The ququard implementation realizes ULC as a single-ququard operation.
Appendix B: Optical Linear-combination implementation for universal two-qubit unitary gates
The optical implementation uses Hilbert-space extension and pre-entangled photonic systems to realize universal two-qubit operations with a compact resource structure. It contrasts this approach with circuit- and cluster-state-based optical schemes and identifies a route to higher success probability.
- Optical implementation: The optical scheme uses spatial modes to encode high-dimensional information in path-encoded qubits.The same approach could also encode information in polarization, time-bin or orbital angular momentum.
- Resource comparison: Universal two-qubit processing requires the equivalent of three consecutive entangling gates in the standard circuit model.The corresponding circuit also requires eight single-qubit gates.
- Resource comparison: Two-photon CNOT protocols can have success probabilities of 1/9 but require coincidence post-selection and therefore cannot implement cascaded CNOT gates.A four-photon protocol reaches 1/4 success probability, according to the comparison.
- Resource comparison: A six-photon cluster state supports at least three two-qubit entangling operations in measurement-based photonic quantum computing.The cited comparison describes this cluster-state resource for MBQC.
- Higher-success design: The current chip’s two-qubit unitary operation has success probability 1/64, with a proposed design improving it to 1/4.The improvement depends on certain signal-idler separation and use of unused optical ports.
Appendix C: Details of Device and Experimental Setup
The device is a silicon-on-insulator photonic chip driven by amplified continuous-wave light and fiber-coupled to external filtering and photon-detection equipment. Signal and idler photons are separated off-chip before coincidence measurements.
- Chip fabrication: The chip uses 220 nm crystalline silicon on a 2 µm buried-oxide silicon-on-insulator substrate.Its 500-nm-wide waveguides are patterned in the silicon layer, with TiN resistive heaters serving as thermo-optic phase shifters.
- Optical setup: A tunable 1550.8 nm continuous-wave laser is amplified by an EDFA and injected through a 48-channel fiber array.The amplified light reaches up to 300 mW before coupling into the chip.
- Optical setup: Two off-chip DWDM filters separate signal and idler photons using channels symmetrically offset from the pump.The selected wavelengths are 1544.2 nm for the signal and 1557.4 nm for the idler.
- System losses: The optical path includes signal and idler filter insertion losses of 1.89 dB and 2.88 dB, respectively, plus 13 dB total coupling loss.The DWDM filters provide at least 45 dB isolation for non-adjacent channels.
- Photon detection: The detection path uses two fiber-coupled SNSPDs, polarization controllers, and a 450 ps coincidence-integration window.The detectors achieve detection efficiency up to 50%.
1. Electrical characteristics of the phase shifters
The phase-shifter calibration maps electrical drive to optical phase response across the chip. Measurements characterize resistance, interference behavior and the distinct current-to-phase relationship of thermo-optic heaters.
- Calibration: The calibration task is to determine each phase shifter’s optical phase response for applied voltage or current.All 62 thermal-optic phase shifters are controlled through a 62-channel current-output system.
- Electrical characterization: The example I-V curve is linear, allowing resistance and voltage-offset parameters to be fitted.The measured resistance is used to relate applied current to heater voltage.
- Electrical characterization: Most phase shifters have resistance around 800 Ω, while the four pump-filter phase shifters are around 580 Ω.The resistance values come from fitted I-V curves.
- Phase response: Thermo-optic phase shifts vary linearly with electrical power but nonlinearly with current.The phase-current response is parameterized by θ(I), with φ1 and φ0 describing the particular phase shifter.
- Phase-shifter classes: The 62 heaters are divided into independent and cascaded phase shifters.Independent devices use directly accessible MZI inputs and outputs, with measured extinction ratios up to 30 dB.
2. Calibrating independent phase shifters
Independent phase shifters are calibrated from single-device interference fringes, whereas cascaded arrays require multidimensional fitting to make all elements reconfigurable.
- Independent phase shifters: A single phase shifter is calibrated by scanning heater current while measuring an MZI output-intensity fringe.The current is swept from 0 to 9 mA in 0.05 mA steps.
- Independent phase shifters: Nonlinear fitting of the measured fringe yields phase parameters φ1 = 0.1123 and φ0 = 0.3814.
- Independent phase shifters: The calibrated independent phase shifters control filters, source intensities, measurement, source phases, and linear-combination stages.
- Cascaded phase-shifter arrays: Five cascaded phase shifters in each array cannot be calibrated as isolated devices because only the array input and output ports are accessible.
- Cascaded phase-shifter arrays: A 5D fringe scan and multiparameter least-squares fitting estimate the array response and make each phase shifter fully reconfigurable.The fitting uses measured output intensities for every five-current configuration and obtains 16 response parameters.
- Cascaded phase-shifter arrays: The 5D calibration method may become challenging for larger cascaded arrays because its data requirement increases exponentially.
4. Calibrating on-chip filters
On-chip pump filters use imbalanced MZIs to suppress pump wavelengths while transmitting signal and idler photons into the functional circuit.
- The on-chip pump filter is implemented as an imbalanced Mach–Zehnder interferometer.
- The filter has low transmission near 1550.8 nm and high transmission near 1544.2 nm and 1557.4 nm for signal and idler photons.
- An extinction ratio of up to ∼28 dB suppresses pump light after the photon sources.
- With up to 300 mW pump power per SFWM source, ∼0.475 mW remains after filtering and adds negligible counts in the functional area.
1. Implementing two-qubit quantum logic gates
The chip implements two-qubit gates through linear combinations of tensor-product operations, including gates requiring three entangling gates in the standard circuit model.
- SWAP is conventionally decomposed into three CNOT gates, while iSWAP combines a unitary operation with SWAP.
- The device implements two-qubit gates by linear combinations of tensor products of Pauli and identity gates.
- SWAP, iSWAP, and √SWAP are implemented using tailored path-entangled states and pump amplitudes or phases.SWAP uses zero relative phase, iSWAP uses π/2 relative phase, and √SWAP requires unbalanced pumps.
- The measured process fidelities for SWAP, iSWAP, and √SWAP are 95.33±0.24%, 94.45±0.27%, and 92.41±0.33%, respectively.
- The process fidelity is calculated as FP = Tr(χidealχexp), with uncertainty estimated by Monte Carlo sampling under Poissonian photon statistics.
3. Demonstrating QAOA
The experiments apply p = 1 QAOA to three constraint-satisfaction problems, identifying optimal or degenerate strings from measured computational-basis outputs.
- CSP1: For CSP1, the objective values are 3, 1, 1, and -1, making {1, 1} the target string.
- QAOA procedure: QAOA searches γ and β to maximize ⟨γ, β|C|γ, β⟩, then measures the corresponding state in the computational basis.
- CSP1: For CSP1, p = 1 QAOA outputs {z1, z2} = {1, 1} with highest probability.
- CSP3: For CSP3, three strings maximize C, and p = 1 QAOA outputs them with approximate probability 1/3 each.
- Experimental cost: Each example uses one two-qubit entangling gate, while evaluating all 600 angle pairs experimentally takes more than 10 hours.
4. Simulating Szegedy quantum walk
The paper analyzes Szegedy quantum-walk periodicity through the eigenvalues of the single-step evolution operator and experimentally simulates walks for varied weights and initial states. The experiments compare measured node probabilities with theoretical distributions over the first 200 steps.
- Theoretical analysis: The walk state after t steps is determined by repeated application of the single-step evolution operator to the initial state.The analysis first decomposes the operator spectrally and expresses the initial state in its eigenbasis.
- Theoretical analysis: The single-step evolution operator is periodic exactly when all eigenvalues are roots of unity sharing a common period.The period is the lowest common multiple of the individual eigenvalue periods.
- Theoretical analysis: For the analyzed two-node cases, specific parameter choices produce periodic walks with periods of 6, 4, 6, and 2 steps, while other cases have periods of 4 and 6 steps.
- Experimental simulation: The experiments implemented walks for varied weights α and β, three initial states, and the first 200 steps, measuring the probability of finding the walker at node 1.
- Experimental simulation: General symmetric weights do not yield perfect periodicity on the example two-node graph, although some cases display quasi-periodic probability distributions.Measured distributions were compared with theoretical distributions, with average fidelities reported in Table I and example traces shown in Fig. S9.