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arXiv:2505.24212

Optimal Haar random fermionic linear optics circuits

Paolo Braccia, N. L. Diaz, Martín Larocca, Marco Cerezo, Diego García-Martín

Published May 30, 2025• 0 citations
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Abstract

Sampling unitary Fermionic Linear Optics (FLO), or matchgate circuits, has become a fundamental tool in quantum information. Such capability enables a large number of applications ranging from randomized benchmarking of continuous gate sets, to fermionic classical shadows. In this work, we introduce optimal algorithms to sample over the non-particle-preserving (active) and particle-preserving (passive) FLO Haar measures. In particular, we provide appropriate distributions for the gates of $n$-qubit parametrized circuits which produce random active and passive FLO. In contrast to previous approaches, which either incur classical $\mathcal{O}(n^3)$ compilation costs or have suboptimal depths, our methods directly output circuits which simultaneously achieve an optimal down-to-the-constant-factor $Θ(n)$ depth and $Θ(n^2)$ gate count; with only a $Θ(n^2)$ classical overhead. Finally, we also provide quantum circuits to sample Clifford FLO with an optimal $Θ(n^2)$ gate count.

DOI
10.48550/arxiv.2505.24212

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Sampling unitary Fermionic Linear Optics (FLO), or matchgate circuits, has become a fundamental tool in quantum information. Such capability enables a large number of applications ranging from randomized benchmarking of continuous gate sets, to fermionic classical shadows. In this work, we introduce optimal algorithms to sample over the non-particle-preserving (active) and particle-preserving (passive) FLO Haar measures. In particular, we provide appropriate distributions for the gates of nnn-qubit parametrized circuits which produce random active and passive FLO. In contrast to previous approaches, which either incur classical O(n3)\mathcal{O}(n^3)O(n3) compilation costs or have suboptimal depths, our methods directly output circuits which simultaneously achieve an optimal down-to-the-constant-factor Θ(n)Θ(n)Θ(n) depth and Θ(n2)Θ(n^2)Θ(n2) gate count; with only a Θ(n2)Θ(n^2)Θ(n2) classical overhead. Finally, we also provide quantum circuits to sample Clifford FLO with an optimal Θ(n2)Θ(n^2)Θ(n2) gate count.
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Fermionic averaged circuit eigenvalue sampling (FACES) is a protocol to simultaneously learn the averaged error rates of many fermionic linear optical (FLO) gates simultaneously and self-consistently from a suitable collection of FLO circuits. It is highly flexible, allowing for the in situ characterization of FLO-averaged gate-dependent noise under natural assumptions on a family of continuously parameterized one- and two-qubit gates. We rigorously show that our protocol has an efficient sampling complexity, owing in-part to useful properties of the Kravchuk transformations that feature in our analysis. We support our conclusions with numerical results. As FLO circuits become universal with access to certain resource states, we expect our results to inform noise characterization and error mitigation techniques on universal quantum computing architectures which naturally admit a fermionic description.
Sampling problems give some of the clearest conditional separations between quantum and classical computation. In Fermion Sampling, passive linear optics acts on a non-Gaussian magic input. Globally Haar-random transformations give anticoncentration and high-precision average-case hardness, but require linear depth and quadratically many gates. A natural open question is whether linear depth is necessary? We show that logarithmic depth suffices for both guarantees in the same ensemble. For fresh uniform matchings with independent Haar two-mode gates, the collision reaches the passive-Haar scale at a sharp threshold of log⁡n/log⁡(9/4)≈0.855log⁡2n\log n/\log(9/4)\approx0.855\log_2 nlogn/log(9/4)≈0.855log2​n layers, with an explicit limiting transition profile and a matching lower bound from two-particle correlations. The mechanism is input-dependent: the magic input suppresses the extensive weight of the slowest relaxation mode, leaving the next to set the scale. An occupation-basis input retains large collision even under full Haar randomness. At a larger logarithmic depth, we prove, in the real-RAM model, average-case #P\#\mathsf{P}#P-hardness of estimating output probabilities to additive error 2−O(nlog⁡2n)2^{-O(n\log^2 n)}2−O(nlog2n) on any fixed fraction of instances above 3/43/43/4. We construct hard instances in four native layers, embed them in typical schedules, and interpolate along Cayley paths using a rational linear-program decoder. A finite 192192192-gate alphabet preserves the collision law exactly. Both guarantees use logarithmic native fermionic depth between arbitrary mode pairs and O(nlog⁡n)O(n\log n)O(nlogn) gates, replacing the global-Haar construction's quadratic budget. The accuracy proved is finer than the 1/N1/N1/N scale, where N=(nn/2)N=\binom{n}{n/2}N=(n/2n​), required by sampling-to-counting; hardness of sampling within constant total-variation distance remains open.
We revisit the problem of learning fermionic linear optics (FLO), also known as fermionic Gaussian unitaries. Given black-box query access to an unknown FLO, previous proposals required O~(n5/ε2)\widetilde{\mathcal{O}}(n^5 / \varepsilon^2)O(n5/ε2) queries, where nnn is the system size and ε\varepsilonε is the error in diamond distance. These algorithms also use unphysical operations (i.e., violating fermionic superselection rules) and/or nnn auxiliary modes to prepare Choi states of the FLO. In this work, we establish efficient and experimentally friendly protocols that obey superselection, use minimal ancilla (at most 111 extra mode), and exhibit improved dependence on both parameters nnn and ε\varepsilonε. For arbitrary (active) FLOs this algorithm makes at most O~(n4/ε)\widetilde{\mathcal{O}}(n^4 / \varepsilon)O(n4/ε) queries, while for number-conserving (passive) FLOs we show that O(n3/ε)\mathcal{O}(n^3 / \varepsilon)O(n3/ε) queries suffice. The complexity of the active case can be further reduced to O~(n3/ε)\widetilde{\mathcal{O}}(n^3 / \varepsilon)O(n3/ε) at the cost of using nnn ancilla. This marks the first FLO learning algorithm that attains Heisenberg scaling in precision. As a side result, we also demonstrate an improved copy complexity of O~(nη2/ε2)\widetilde{\mathcal{O}}(n η^2 / \varepsilon^2)O(nη2/ε2) for time-efficient state tomography of ηηη-particle Slater determinants in ε\varepsilonε trace distance, which may be of independent interest.
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.
Fermionic Gaussian states (FGSs) and the associated matchgate circuits play a central role in quantum information theory and condensed matter physics. Despite being possibly highly entangled, they can still be efficiently simulated on classical computers. We address the question of how to optimally create such states when using matchgate circuits acting on product states. To this end, we derive lower bounds on the number of gates required to prepare an arbitrary pure FGS: We establish both an asymptotic bound on the minimal gate count over general nearest-neighbor gate sets and an exact bound for circuits composed solely of matchgates. We present explicit algorithms whose constructions saturate these bounds, thereby proving their optimality. We furthermore determine when an FGS can be prepared with a circuit of any given depth, and derive an algorithm that constructs such a circuit whenever this condition is satisfied, either exactly or approximately. Our results have direct applications to (approximate) state preparation and to disentangling procedures. Moreover, we introduce a new classical simulation algorithm for matchgate circuits, based entirely on manipulating the generating circuits of the FGSs. Finally, we briefly study an extension of our framework for ttt-doped Gaussian states and circuits.