The experimental interest and developments in quantum spin-1/2 chains has increased uninterruptedly over the past decade. In many instances, the target quantum simulation belongs to the broader class of noninteracting fermionic models, constituting an important benchmark. In spite of this class being analytically efficiently tractable, no direct certification tool has yet been reported for it. In fact, in experiments, certification has almost exclusively relied on notions of quantum state tomography scaling very unfavorably with the system size. Here, we develop experimentally friendly fidelity witnesses for all pure fermionic Gaussian target states. Their expectation value yields a tight lower bound to the fidelity and can be measured efficiently. We derive witnesses in full generality in the Majorana-fermion representation and apply them to experimentally relevant spin-1/2 chains. Among others, we show how to efficiently certify strongly out-of-equilibrium dynamics in critical Ising chains. At the heart of the measurement scheme is a variant of importance sampling specially tailored to overlaps between covariance matrices. The method is shown to be robust against finite experimental-state infidelities.
Fermionic Gaussian states are the workhorse reference states for qubit-based quantum simulation of fermionic systems, yet existing certification protocols either proceed via fidelity estimation, leading to suboptimal sample complexity in the target precision
ε, only apply to Haar-typical states, or require adaptive measurements on a large fraction of qubits. We give an adaptive protocol that certifies any
d-mode pure fermionic Gaussian state using
O(d2ε−1) copies, single-qubit measurements with only one qubit measured adaptively per copy, and
O(d3) classical processing time per copy. The
ε−1 dependence is optimal for the certification problem, even among strategies using entangled measurements. The sample complexity is controlled by the spectral gap of a
d↔d−2 down-up walk---a Markov chain studied in the theory of high-dimensional expanders, equivalent to a two-site Glauber dynamics in statistical physics---relating certification efficiency to relaxation time to equilibrium. The worst-case bound is tight for this protocol and is attained by physically relevant states, including ground states of the fully dimerized Su-Schrieffer-Heeger (SSH) chain and BCS pair states. In contrast, numerics up to
d=14 suggest that
O(dε−1) copies suffice to certify Haar-typical Gaussian states, a factor of
d improvement over the worst-case bound. Finally, because the bound depends only on the target's computational-basis distribution, the protocol extends to efficiently phase-dressed states---states obtained by injecting an efficiently computable diagonal phase to Gaussian states---with the same sample complexity. The class includes a continuous family of four-mode non-Gaussian magic states for matchgate computation.
Fermionic Gaussian states describe free fermions and underlie the mean-field picture of matter, from metals to superconductors; they are also efficiently simulable on classical computers. Departures from Gaussianity -- the correlations produced by interactions -- are therefore what make a fermionic system hard to simulate classically and useful for quantum computation, analogous to the role of magic in stabilizer-based quantum computation. Yet detecting and quantifying such non-Gaussianity at scale has remained challenging. Here we introduce practical tests and witnesses of fermionic non-Gaussianity built on fermionic antiflatness, a measure derived from the two-point covariance matrix. We estimate it with two protocols -- a two-copy Bell measurement and a single-copy scheme using commuting Majorana bilinears -- that determine whether a state is Gaussian or far from it at lower measurement cost than existing approaches, using only operations native to fault-tolerant hardware. For mixed states, a purity-corrected witness certifies non-Gaussianity and remains robust under strong noise; running it on the IQM quantum processor, we find that noise can both reduce and enhance non-Gaussianity. Finally, we show that preparing pseudorandom fermionic states requires extensive non-Gaussianity. Together, these tools enable the study and certification of non-Gaussian fermionic resources on present-day quantum devices.
Interacting fermions owe both their non-trivial correlations and their computational power to their departure from Gaussianity. Yet, quantifying that departure geometrically faces two challenges: the Gaussian manifold is non-convex, and perturbative treatments are confined to weak interactions. Here, we introduce a geometric measure of fermionic non-Gaussianity, the fidelity distance of a target state to the manifold of fermionic Gaussian states. We construct an iteration formalism to numerically access it, applicable also at strong coupling. Further, we prove a sharpened operator Jensen inequality, of interest in its own right. We use it to derive a witness that involves only two- and four-point fermionic correlators. Numerical benchmarks on the one-dimensional Fermi--Hubbard model show that the quality of the witness improves with system size and, for strong coupling, is tighter than antiflatness. Our results open a scalable route to charting non-Gaussianity of strongly-interacting fermionic quantum simulators.
Quantum memories are key ingredients for future quantum networks. Developing methods for their certification is therefore an important, yet challenging, problem. In this work, we present a general framework for certifying quantum memories and quantifying their performance in an operational and efficient manner, taking advantage of developments in the theory of quantum channels. We start by giving a pedagogical introduction to these concepts and tools and illustrate their natural application to quantum memories. Specifically, we quantify the fidelity of a quantum memory via the notion of average channel fidelity, which can be estimated precisely in an experiment via a finite set of probe states forming a so-called
2-design. We then present a number of witnessing techniques for certifying this quantity, and show that it gives access to other important properties of the channel, such as its dimensionality and its capacity to transmit quantum information. We consider mostly a simple prepare-and-measure scenario, but discuss also the case of an entanglement-based setup, as well as a semi-device-independent approach. Finally, we apply these methods to a solid-state quantum memory experiment, certifying an average channel fidelity of 98.6%. Our certification methods can be readily applied to other quantum devices, such as fibers, transducers and gates.
Majorana fermions arising in superconducting nanowires are of interest both for fundamental reasons and due to their potential use as building blocks for a topological quantum computer. While recent experiments have reported strong evidence of Majorana fermions in the form of occupation of a zero-energy fermionic mode, there is still disagreement about whether the observed data can be explained by other mechanisms. Furthermore, to be useful for quantum computation, the fermionic parities of zero-energy modes must be measured with high fidelity. In this thesis, we apply a quantum information protocol known as self-testing to a system of Majorana fermions. A self-testing protocol is a statistical test for certifying a quantum state and a set of measurement operators from experimentally determined expectation values, under minimal physical assumptions. Such a test treats the system as a black-box and relies only on the distributions of measurement outcomes given device settings. For the protocol described in this thesis, if the test is passed, then a set of measurements that ideally are of fermionic parities are certified to have the exact anticommutation relations as determined by the anticommutativity of Majorana fermion operators. The protocol is robust in the sense that bounds on the state and operator fidelities can still be established in the presence of small experimental imperfections. Finally, a rigorous statistical analysis is used to derive a confidence interval for the certified fidelities given a finite experimental data set.
We introduce witnesses for the average channel fidelity between a known target gate and an arbitrary unknown channel, for continuous-variable (CV) systems. These are observables whose expectation value yields a tight lower bound to the average channel fidelity in question, thus constituting a practical tool for benchmarking experimental CV gates. Our framework applies to a broad class of target gates. Here, we focus on three specific types of targets: multi-mode Gaussian unitary channels, single-mode coherent state amplifiers, and the single-mode (non-Gaussian) cubic phase gate, which is a crucial ingredient for CV universal quantum computation. Our witnesses are experimentally-friendly as they rely exclusively on Gaussian measurements, even for the non-Gaussian-target case. Moreover, in all three cases, they can be measured efficiently in the estimation error
ϵ and failure probability
Δ, as well as in the number of modes
m for the Gaussian-target case. To end up with, our approach for the Gaussian-target case relies on an improved measurement scheme for Gaussian state-fidelity witnesses, which is polynomially and exponentially more efficient in
m and
Δ, respectively, than previous schemes. The latter constitutes an interesting byproduct result on its own. Our findings are relevant to the experimental validation of many-body quantum technologies.