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Experimental Bayesian Quantum Phase Estimation on a Silicon Photonic Chip
Stefano Paesani, Andreas A. Gentile, Raffaele Santagati, Jianwei Wang, Nathan Wiebe, David P. Tew, Jeremy L. O'Brien, Mark G. Thompson
TL;DR
Quantum phase estimation is important for algorithms such as factorization and molecular simulation, but its practicality on near-term hardware remains uncertain. This work implements Bayesian RFPE on a silicon photonic device and compares it with IPEA under experimental imperfections, finding evidence that RFPE is robust to realistic errors while the photonic implementation uses a post-selected scheme that is not scalable.
Problem
Quantum phase estimation is a key subroutine for factorization and molecular simulation, but its practical viability on near-term quantum hardware requires experimental evidence under realistic noise.
Method
The paper implements rejection filtering phase estimation on a silicon photonic device and compares it with iterative phase estimation using photonic measurements and controlled experimental imperfections.
Results
The experiments provide evidence that RFPE is more robust to realistic errors than IPEA, while phase-shifter calibration fits achieve R^2 values close to one.
Takeaways & Limitations
The results support RFPE as a promising approach for experimental near-term quantum phase estimation, although the demonstrated controlled-unitary implementation is not scalable because it is post-selected.
Takeaways & Limitations
The controlled-unitary implementation is post-selected and therefore not scalable.
Abstract
from arXiv · showhide
Quantum phase estimation is a fundamental subroutine in many quantum algorithms, including Shor's factorization algorithm and quantum simulation. However, so far results have cast doubt on its practicability for near-term, non-fault tolerant, quantum devices. Here we report experimental results demonstrating that this intuition need not be true. We implement a recently proposed adaptive Bayesian approach to quantum phase estimation and use it to simulate molecular energies on a Silicon quantum photonic device. The approach is verified to be well suited for pre-threshold quantum processors by investigating its superior robustness to noise and decoherence compared to the iterative phase estimation algorithm. This shows a promising route to unlock the power of quantum phase estimation much sooner than previously believed.
Appendix A: Iterative Phase Estimation Algorithm (IPEA)
IPEA iteratively infers eigenphase bits, beginning with the least significant digit and increasing precision through controlled experiments. Experimental noise can accumulate because an incorrect bit cannot be corrected later.
- IPEA infers each eigenphase digit iteratively, starting with the least significant bit and proceeding toward more significant bits.
- For n-bit precision, the j-th iteration uses M_j = 2^(n−j), with the initial step setting θ_1 = 0 and M_1 = 2^(n−1).
- Noise sources considered for IPEA include dephasing with decoherence time T_2 and control-qubit R_x(δ) errors with Gaussian angle variance Δ_x.
- Majority voting can increase the probability of a correct bit measurement, but the protocol retains exponential scaling with noise and cannot recover after an inference error.
Appendix B: Rejection Filtering Phase Estimation (RFPE)
RFPE represents phase uncertainty with a prior distribution, adaptively chooses experiments, and updates the distribution using Bayesian rejection filtering. This avoids storing a finely discretized distribution, whose memory demands can become impractical.
- RFPE uses the same experimental setup as traditional iterative phase estimation but differs in adaptive experiment selection and data processing.
- The prior distribution P(φ) represents confidence in candidate eigenphases and is updated into a posterior using observed evidence and the likelihood function.
- A discretized Bayesian prior requiring error on the order of 10^-9 can need several gigabytes of memory, while multiple eigenvalue hypotheses increase storage exponentially.
- Rejection filtering samples particles from a Gaussian prior, accepts them probabilistically according to P(E|x), and estimates posterior mean and variance from accepted particles.
- The Particle Guess Heuristic chooses M = ⌈1.25/σ⌉ and θ ∼ N(µ, σ) to obtain near-optimal experiment parameters without precomputation.
- RFPE iterates experiment, measurement, and Bayesian-update steps from an initial N(µ_0, σ_0), returning the estimated phase µ and uncertainty σ.
1. The silicon device and experimental details
The experiment uses a silicon-on-insulator photonic chip with integrated photon sources, path-encoded qubits, thermo-optic control, and active temperature and coupling stabilization. These components provide the optical platform for implementing the phase-estimation experiments.
- The device is a silicon-on-insulator chip fabricated by deep-UV photolithography and dry etching, with fiber coupling through inverse-taper spot-size converters.
- A Peltier device with PID control stabilizes chip temperature, while optical fibers are automatically recoupled before each scan to maximize coupling efficiency.
- Two spiral waveguide sources generate entangled photon pairs through spontaneous four-wave mixing using silicon’s intrinsic χ^(3) nonlinearity.
- The qubits encode information in photon paths, with signal and idler wavelengths selected at 1545.5 nm and 1558.3 nm, respectively.
- Thermo-optic phase shifters use independently controlled resistive heaters to change the refractive index and implement reconfigurable quantum operations.
2. State evolution and scheme for arbitrary controlled-unitaries
The device realizes arbitrary controlled-unitary operations by combining entangled photon paths, single-qubit gates, path erasure, and post-selection. The resulting operation is experimentally useful but not scalable because the scheme is post-selected.
- 2. State evolution and scheme for arbitrary controlled-unitaries: The controlled-unitary circuit requires two-qubit control, implemented using an entanglement-based scheme with coherently pumped spiral SFWM sources.
- 2. State evolution and scheme for arbitrary controlled-unitaries: Post-selection on signal and idler output modes prepares the path-encoded photonic state used for the controlled operation.
- 2. State evolution and scheme for arbitrary controlled-unitaries: An additional target-path degree of freedom creates an entangled control-target-path state before gate operations are applied.
- 2. State evolution and scheme for arbitrary controlled-unitaries: Waveguide crossings and MMI beam splitters erase path information, producing a superposition of identity and U operations on the target register.
- 2. State evolution and scheme for arbitrary controlled-unitaries: Projecting the third qubit into |0⟩_P yields a state equivalent to applying the desired arbitrary controlled-unitary operation.
- 2. State evolution and scheme for arbitrary controlled-unitaries: The implementation is post-selected and therefore not scalable, although it enables experiments previously inaccessible to integrated quantum photonics.
Appendix D: Estimating and simulating phase errors in the integrated photonic device
The device uses calibrated thermo-optical phase shifters, with independent current control and compensation for thermal cross-talk, to implement target phases. Calibration fits support the device model and quantify phase uncertainty.
- Device phase control: Thermo-optical heaters change waveguide refractive indices to control the implemented phase shifts.Each heater is independently driven, while shared-ground and thermal cross-talk can introduce unwanted changes in other optical paths.
- Calibration model: The calibration model fits output-power oscillations as a nonlinear function of heater power.The fit parameters include background B, maximum amplitude A, period T, and offset power PΦ.
- Calibration model: The fitted model adequately reproduces measured device behavior, with R2 values close to one across reported calibrations.The analysis also reports high t-statistics and low p-values for most fitted parameters.
- Phase uncertainty: The target phase is set by driving each heater at a calibrated electrical power, while uncertainties in Pel, T, and PΦ propagate into the phase.Driver-current inaccuracies are below 0.04% for all heaters in the standard configuration.
- Phase uncertainty: The experimentally estimated average phase precision is σexp ≃0.01 rad after accounting for uncertainties in T and PΦ.The calibration procedure substantially reduces systematic errors, although intrinsic stochastic phase uncertainty remains.
Appendix E: Photonic implementation of RFPE
The photonic implementation must convert bulk photon-count measurements into data suitable for RFPE. Majority voting is usable for this comparison, but sampled measurements can retain more information and expose robustness limitations.
- Measurement strategies: Photonic experiments return batches of measurement samples rather than the single-shot data assumed by the original RFPE formulation.The appendix therefore compares majority voting with random sampling from measured output statistics.
- Measurement strategies: Majority voting converts bulk qubit counts into one datum by selecting the more frequent outcome.This procedure uses only part of the information contained in the bulk measurement.
- Performance comparison: Majority voting outperforms N = 1 RFPE, requiring polynomially fewer measurements to reach the same fixed accuracy.Reduced measurement uncertainty benefits the RFPE inference process in this comparison.
- Robustness and limitations: Majority voting gives qualitatively similar results while only partially exploiting the enhanced information in photonic measurements.The authors use it here for comparison with IPEA, while noting that likelihoods tailored to bulk measurements could improve RFPE.
Appendix F: Analysis of Breakdown of Majority Voting in IPEA
The analysis explains IPEA’s rapid breakdown under noise as a consequence of majority voting combined with its inability to detect and correct inferred-bit errors. Error suppression weakens as measurement noise grows, while the required shots increase with the number of inferred bits.
- Noise threshold: As Pe approaches 1, the Chernoff-bound exponent tends to zero, eliminating the exponential error suppression normally gained from majority voting.For small Pe, the exponent is approximately 1.
- Noise threshold: Exponential error suppression from majority voting holds when Pe is small and P0 remains bounded above 1/2.
- Noise threshold: 500 shots with P0 = 2/3 substantially suppress bit-inference errors until Pe ≈1/3, after which the error probability rapidly diverges.
- Scaling with bit count: As the number of inferred bits increases, the number of shots needed per bit must also increase, although only modestly relative to the bit count.
- Interpretation: The rapid IPEA breakdown follows from majority voting and its inability to detect or adapt after an inference error occurs.
Appendix G: Rescaling the decoherence time T2
The decoherence-time parameter T2 is rescaled into architecture-dependent physical time using the duration of each controlled-U operation. This translation shows that scalable IPEA can become impractical when hardware decoherence times are too short.
- Definition and scaling: T2 measures decoherence time in units of the time required for each controlled gate contributing to controlled-U^M, so its physical value depends on the architecture.
- Architecture-dependent implications: For superconductive devices, a scalable 16-bit IPEA implementation becomes impractical below approximately 50 µs decoherence time.The estimate uses an approximately 1.5 µs operational time for a three-qubit controlled-U.
- Architecture-dependent implications: A solid-state electron-spin implementation would require approximately 1.5 ms total time for a scalable 16-bit IPEA, compared with typically tens-of-milliseconds decoherence times.