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arXiv:1809.04680 · Physical Review A

Production of photonic universal quantum gates enhanced by machine learning

Krishna Kumar Sabapathy, Haoyu Qi, Josh Izaac, Christian Weedbrook

Published Jul 17, 2019• 62 citations
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Abstract

We introduce photonic architectures for universal quantum computation. The first step is to produce a resource state which is a superposition of the first four Fock states with a probability greater than or equal to ${10}^{\ensuremath{-}2}$, an increase by a factor of ${10}^{4}$ over standard sequential photon-subtraction techniques. The resource state is produced with near-perfect fidelity from a quantum gadget that uses displaced squeezed vacuum states, interferometers, and photon-number-resolving detectors. The parameters of this gadget are trained using machine learning algorithms for variational circuits. We discuss in detail various aspects of the non-Gaussian state preparation resulting from the numerical experiments. We then propose a notion of resource farms where these gadgets are stacked in parallel, to increase the success probability further. We find a trade-off between the success probability of the farm, the error tolerance, and the number of gadgets. Using the resource states in conventional gate teleportation techniques, we can then implement weak tunable cubic phase gates. The numerical tools that have been developed could potentially be useful for other applications in photonics as well.

DOI
10.1103/physreva.100.012326

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We introduce photonic architectures for universal quantum computation. The first step is to produce a resource state which is a superposition of the first four Fock states with a probability greater than or equal to 10\ensuremath−2{10}^{\ensuremath{-}2}10\ensuremath−2, an increase by a factor of 104{10}^{4}104 over standard sequential photon-subtraction techniques. The resource state is produced with near-perfect fidelity from a quantum gadget that uses displaced squeezed vacuum states, interferometers, and photon-number-resolving detectors. The parameters of this gadget are trained using machine learning algorithms for variational circuits. We discuss in detail various aspects of the non-Gaussian state preparation resulting from the numerical experiments. We then propose a notion of resource farms where these gadgets are stacked in parallel, to increase the success probability further. We find a trade-off between the success probability of the farm, the error tolerance, and the number of gadgets. Using the resource states in conventional gate teleportation techniques, we can then implement weak tunable cubic phase gates. The numerical tools that have been developed could potentially be useful for other applications in photonics as well.
We introduce architectures for near-deterministic implementation of fully tunable weak cubic phase gates requisite for universal quantum computation. The first step is to produce a resource state which is a superposition of the first four Fock states with a probability ≥10−2\geq 10^{-2}≥10−2, an increase by a factor of 10410^4104 over standard sequential photon-subtraction techniques. The resource state is produced from a quantum gadget that uses displaced squeezed states, interferometers and photon-number resolving detectors. The parameters of this gadget are trained using machine learning algorithms for variational circuits. Stacking these gadgets in parallel we build quantum resource farms in a scalable manner depending on the error tolerance. Using conventional teleportation techniques we can implement weak cubic phase gates, in principle, at a rate ∼100kHz\sim {\rm 100 kHz}∼100kHz dictated by the photon number resolving detectors. Our proposal is realizable with current photonic technologies without the need for quantum memories. The methods for non-Gaussian state preparation is of independent interest to the resource theory of non-Gaussianity.
Universal quantum computation using photonic systems requires gates the Hamiltonians of which are of order greater than quadratic in the quadrature operators. We first review previous proposals to implement such gates, where specific non-Gaussian states are used as resources in conjunction with entangling gates such as the continuous-variable versions of controlled-phase and controlled-not gates. We then propose ON states which are superpositions of the vacuum and the NthN\mathrm{th}Nth Fock state, for use as non-Gaussian resource states. We show that ON states can be used to implement the cubic and higher-order quadrature phase gates to first order in gate strength. There are several advantages to this method such as reduced number of superpositions in the resource state preparation and greater control over the final gate. We also introduce useful figures of merit to characterize gate performance. Utilizing a supply of on-demand resource states one can potentially scale up implementation to greater accuracy, by repeated application of the basic circuit.
Cubic-phase states are a sufficient gate resource for universal quantum computing over continuous variables. We present results from numerical experiments in which deep neural networks are trained via reinforcement learning to control a quantum optical circuit for generating cubic-phase states by leveraging quantum interference in phase space, with an unprecedented average success rate of 96 % . The only non-Gaussian resource required is photon-number-resolving measurements. We also show that the exact same resources enable the direct generation of a quartic-phase gate, with no need for a cubic gate decomposition.
We present numerical results from simulations using deep reinforcement learning to control a measurement-based quantum processor—a time-multiplexed optical circuit sampled by photon-number-resolving detection—and find it generates squeezed cat states quasi-deterministically, with an average success rate of 98%, far outperforming all other proposals. Since squeezed cat states are deterministic precursors to the Gottesman–Kitaev–Preskill (GKP) bosonic error code, this is a key result for enabling fault tolerant photonic quantum computing. Informed by these simulations, we also discovered a one-step quantum circuit of constant parameters that can generate GKP states with high probability, though not deterministically.
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.
We present a classical algorithm for simulating universal quantum circuits composed of "free" nearest-neighbour matchgates or equivalently fermionic-linear-optical (FLO) gates, and "resourceful" non-Gaussian gates. We achieve the promotion of the efficiently simulable FLO subtheory to universal quantum computation by gadgetizing controlled phase gates with arbitrary phases employing non-Gaussian resource states. Our key contribution is the development of a novel phase-sensitive algorithm for simulating FLO circuits. This allows us to decompose the resource states arising from gadgetization into free states at the level of statevectors rather than density matrices. The runtime cost of our algorithm for estimating the Born-rule probability of a given quantum circuit scales polynomially in all circuit parameters, except for a linear dependence on the newly introduced FLO e x t e n t , which scales exponentially with the number of controlled-phase gates. More precisely, as a result of finding optimal decompositions of relevant resource states, the runtime doubles for every maximally resourceful (e.g., swap or CZ) gate added. Crucially, this cost compares very favourably with the best known prior algorithm, where each swap gate increases the simulation cost by a factor of approximately 9. For a quantum circuit containing arbitrary FLO unitaries and k controlled-Z gates, we obtain an exponential improvement O ( 4.5 k ) over the prior state-of-the-art.