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arXiv:2109.01444 · Physical Review Letters

Efficient Backcasting Search for Optical Quantum State Synthesis

Kosuke Fukui, Shuntaro Takeda, Mamoru Endo, Warit Asavanant, Jun–ichi Yoshikawa, Peter van Loock, Akira Furusawa

Published Jun 17, 2022• 26 citations
arXivHTMLPDF

Abstract

Non-Gaussian states are essential for many optical quantum technologies. The so-called optical quantum state synthesizer (OQSS), consisting of Gaussian input states, linear optics, and photon-number resolving detectors, is a promising method for non-Gaussian state preparation. However, an inevitable and crucial problem is the complexity of the numerical simulation of the state preparation on a classical computer. This problem makes it very challenging to generate important non-Gaussian states required for advanced quantum information processing. Thus, an efficient method to design OQSS circuits is highly desirable. To circumvent the problem, we offer a scheme employing a backcasting approach, where the circuit of OQSS is divided into some sublayers, and we simulate the OQSS backwards from final to first layers. Moreover, our results show that the detected photon number by each detector is at most 2, which can significantly reduce the requirements for the photon-number resolving detector. By virtue of the potential for the preparation of a wide variety of non-Gaussian states, the proposed OQSS can be a key ingredient in general optical quantum information processing.

DOI
10.1103/physrevlett.128.240503

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Non-Gaussian states are essential for many optical quantum technologies. The so-called optical quantum state synthesizer (OQSS), consisting of Gaussian input states, linear optics, and photon-number resolving detectors, is a promising method for non-Gaussian state preparation. However, an inevitable and crucial problem is the complexity of the numerical simulation of the state preparation on a classical computer. This problem makes it very challenging to generate important non-Gaussian states required for advanced quantum information processing. Thus, an efficient method to design OQSS circuits is highly desirable. To circumvent the problem, we offer a scheme employing a backcasting approach, where the circuit of OQSS is divided into some sublayers, and we simulate the OQSS backwards from final to first layers. Moreover, our results show that the detected photon number by each detector is at most 2, which can significantly reduce the requirements for the photon-number resolving detector. By virtue of the potential for the preparation of a wide variety of non-Gaussian states, the proposed OQSS can be a key ingredient in general optical quantum information processing.
Quantum state preparation is important for quantum information processing. In particular, in optical quantum computing with continuous variables, non-Gaussian states are needed for universal operation and error correction. Optical non-Gaussian states are usually generated by heralding schemes using photon detectors. In previous experiments, the temporal resolution of the photon detectors was sufficiently high relative to the time width of the quantum state, so that the conventional theory of non-Gaussian state preparation treated the detector's temporal resolution as negligible. However, when using various photon detectors, including photon-number-resolving detectors, the temporal resolution is non-negligible. In this paper, we extend the conventional theory of quantum state preparation using photon detectors to the finite-temporal-resolution regime, analyze the cases of single-photon and two-photon preparation as examples, and find that the generated states are characterized by the dimensionless parameter BBB, defined as the product of the temporal resolution of the detectors $\mathrm{\ensuremath{\Delta}}t$ and the bandwidth of the light source $\mathrm{\ensuremath{\Delta}}f$. Based on the results, $B\ensuremath{\sim}0.1$ is required to keep the purity and fidelity of the generated quantum states high.
Non-Gaussian states of light, which are essential in fault-tolerant and universal optical quantum computation, are typically generated by a heralding scheme using photon detectors. Recently, it has been theoretically shown that the large timing jitter of the photon detectors deteriorates the purity of the generated non-Gaussian states [T. Sonoyama et al., Phys. Rev. A 105, 043714 (2022)]. Here, we circumvent this problem by time-gated photon detection and generate non-Gaussian states with Wigner negativity using continuous-wave light. We use a fast optical switch for time gating to effectively improve the timing jitter of a photon-number-resolving detector based on a transition edge sensor from 50 ns to 10 ns. As a result, we generate non-Gaussian states with Wigner negativity of \ensuremath{-}$0.011\ifmmode\pm\else\textpm\fi{}0.004$ $(\ensuremath{\hbar}=1)$, which cannot be observed without the time-gated photon detection method. These results confirm the effect of the timing jitter on non-Gaussian state generation experimentally for the first time. The proposed time-gated photon detection method would be useful for high-purity non-Gaussian state preparation in optical quantum information processing.
Photonic qubits play an important role in quantum information processing. However, to date, efficient high-speed single photon sources and photon number resolving (PNR) detectors are still difficult to make. Here, we use phase-randomized coherent states and threshold detectors, combined with decoy-state post-processing, to simulate single photon sources and PNR detectors and perform various quantum information processing tasks. We propose a simple linear regression method to predict Fock state statistics based on observable statistics from coherent-state probes, combined with random/structured sampling of probe settings. Compared with traditional decoy-state post-processing based on linear programming and Cartesian product grid sampling, our method enables drastically improved accuracy and much reduced probe resources. It works particularly well for unknown unitary circuits of interest with fixed-size Hilbert space dimensions. For instance, it can accurately simulate 4-photon interference in linear optics with just 32 input/output probe settings and 6-photon interference with 256 probe settings. Moreover, it also works for non-unitary (such as lossy, nonlinear, or arbitrary) quantum channels. This enables a wide variety of applications that can be implemented with easily attainable coherent light sources and threshold detectors, such as characterization of small-scale linear optical quantum computing circuits or quantum sensing/metrology devices.
Linear optical quantum circuits with photon number resolving (PNR) detectors are used for both Gaussian Boson Sampling (GBS) and for the preparation of non-Gaussian states such as Gottesman-Kitaev-Preskill (GKP), cat and NOON states. They are crucial in many schemes of quantum computing and quantum metrology. Classically optimizing circuits with PNR detectors is challenging due to their exponentially large Hilbert space, and quadratically more challenging in the presence of decoherence as state vectors are replaced by density matrices. To tackle this problem, we introduce a family of algorithms that calculate detection probabilities, conditional states (as well as their gradients with respect to circuit parametrizations) with a complexity that is comparable to the noiseless case. As a consequence we can simulate and optimize circuits with twice the number of modes as we could before, using the same resources. More precisely, for an M -mode noisy circuit with detected modes D and undetected modes U , the complexity of our algorithm is O(M2∏i∈UCi2∏i∈DCi) , rather than O(M2∏i∈D∪UCi2) , where Ci is the Fock cutoff of mode i . As a particular case, our approach offers a full quadratic speedup for calculating detection probabilities, as in that case all modes are detected. Finally, these algorithms are implemented and ready to use in the open-source photonic optimization library MrMustard.
Generation of high-fidelity photonic non-Gaussian states is a crucial ingredient for universal quantum computation using continuous-variable platforms, yet it remains a challenge to do this efficiently. We present a general framework for a probabilistic production of multimode non-Gaussian states by measuring a few modes of multimode Gaussian states via photon-number-resolving detectors. We use Gaussian elements consisting of squeezed displaced vacuum states and interferometers, the only non-Gaussian elements consisting of photon-number-resolving detectors. We derive analytic expressions for the output Wigner function, and the probability of generating the states in terms of the mean and the covariance matrix of the Gaussian state and the photon detection pattern. We find that the output states can be written as a Fock-basis superposition state followed by a Gaussian gate, and we derive explicit expressions for these parameters. These analytic expressions show exactly what non-Gaussian states can be generated by this probabilistic scheme. Further, it provides a method to search for the Gaussian circuit and measurement pattern that produce a target non-Gaussian state with optimal fidelity and success probability. We present specific examples such as the generation of cat states, ON states, Gottesman-Kitaev-Preskill states, NOON states, and bosonic-code states. The proposed framework has potentially far-reaching implications for the generation of bosonic error-correction codes that require non-Gaussian states and resource states for the implementation of non-Gaussian gates needed for universal quantum computation, among other applications requiring non-Gaussianity. The tools developed here could also prove useful for the quantum resource theory of non-Gaussianity.
Abstract. Quantum information offers the promise of being able to perform certain communication and computation tasks that cannot be done with conventional information technology (IT). Optical Quantum Information Processing (QIP) holds particular appeal, since it offers the prospect of communicating and computing with the same type of qubit. Linear optical techniques have been shown to be scalable, but the corresponding quantum computing circuits need many auxiliary resources. Here we present an alternative approach to optical QIP, based on the use of weak cross-Kerr nonlinearities and homodyne measurements. We show how this approach provides the fundamental building blocks for highly efficient non-absorbing single photon number resolving detectors, two qubit parity detectors, Bell state measurements and finally near deterministic control-not (CNOT) gates. These are essential QIP devices.