Jonas Helsen

We study the formation of short-depth designs beyond the unitary group. We provide a range of results on several groups of broad interest in quantum information science: the Clifford group, the orthogonal group, the unitary symplectic groups, and the matchgate group. For all of these groups, we prove that analogues of unitary designs cannot be generated by any circuit ensemble with light-cones that are smaller than the system size. This implies linear lower bounds on the circuit depth in one-dimensional systems. For the Clifford, orthogonal, and unitary symplectic group, we moreover show that commonly considered circuit ensembles cannot generate designs in sub-linear depth on any circuit architecture. We show this by exploiting observables in the higher-order commutants of each group, which allow one to distinguish any short-depth circuit from truly random. While these no-go results rule out short-depth designs over these subgroups, we prove that slightly weaker forms of randomness---including additive-error state designs and anti-concentration in sampling distributions---nevertheless emerge at logarithmic depths in many cases. Our results reveal that the onset of randomness in shallow quantum circuits is a widespread yet subtle phenomenon, dependent on the interplay between the group itself and the context of its application.

We consider the problem of Clifford testing, which asks whether a black-box $n$-qubit unitary is a Clifford unitary or at least $\varepsilon$-far from every Clifford unitary. We give the first 4-query Clifford tester, which decides this problem with probability~$\mathrm{poly}(\varepsilon)$. This contrasts with the minimum of 6 copies required for the closely-related task of stabilizer testing. We show that our tester is tolerant, by adapting techniques from tolerant stabilizer testing to our setting. In doing so, we settle in the positive a conjecture of Bu, Gu and Jaffe, by proving a polynomial inverse theorem for a non-commutative Gowers 3-uniformity norm. We also consider the restricted setting of single-copy access, where we give an $O(n)$-query Clifford tester that requires no auxiliary memory qubits or adaptivity. We complement this with a lower bound, proving that any such, potentially adaptive, single-copy algorithm needs at least $\Omega(n^{1/4})$ queries. To obtain our results, we leverage the structure of the commutant of the Clifford group, obtaining several technical statements that may be of independent interest.

We define a general formulation of quantum PCPs, which captures adaptivity and multiple unentangled provers, and give a detailed construction of the quantum reduction to a local Hamiltonian with a constant promise gap. The reduction turns out to be a versatile subroutine to prove properties of quantum PCPs, allowing us to show: (i) Non-adaptive quantum PCPs can simulate adaptive quantum PCPs when the number of proof queries is constant. In fact, this can even be shown to hold when the non-adaptive quantum PCP picks the proof indices simply uniformly at random from a subset of all possible index combinations, answering an open question by Aharonov, Arad, Landau and Vazirani (STOC '09). (ii) If the q-local Hamiltonian problem with constant promise gap can be solved in 𝖰𝖢𝖬𝖠, then 𝖰𝖯𝖢𝖯[q] is in 𝖰𝖢𝖬𝖠 for any constant q. (iii) If 𝖰𝖬𝖠(k) has a quantum PCP for any k=poly(n), then 𝖰𝖬𝖠(2) = 𝖰𝖬𝖠, connecting two of the longest-standing open problems in quantum complexity theory. Moreover, we also show that there exists (quantum) oracles relative to which certain quantum PCP statements are false. Hence, any attempt to prove the quantum PCP conjecture requires, just as was the case for the classical PCP theorem, (quantumly) non-relativizing techniques.

Randomized shadow estimation is a recent protocol that allows estimating exponentially many expectation values of a quantum state from ``classical shadows'', obtained by applying random quantum circuits and computational basis measurements. In this paper we study the statistical efficiency of this approach in light of near-term quantum computing. In particular, we propose and analyze a more practically-implementable variant of the protocol, thrifty shadow estimation, in which quantum circuits are reused many times instead of having to be freshly generated for each measurement (as in the original protocol). We show that the effect of this reuse strongly depends on the family of quantum circuits that is chosen. In particular, it is maximally effective when sampling Haar random unitaries, and maximally ineffective when sampling Clifford circuits (even though the Clifford group forms a three-design). To interpolate between these two extremes, we provide an efficiently simulable family of quantum circuits inspired by a recent construction of approximate t-designs. Finally we consider tail bounds for shadow estimation and discuss when median-of-means estimation can be replaced with standard mean estimation.

We introduce spectral quantum tomography, a simple method to extract the eigenvalues of a noisy few-qubit gate, represented by a trace-preserving superoperator, in a SPAM-resistant fashion, using low resources in terms of gate sequence length. The eigenvalues provide detailed gate information, supplementary to known gate-quality measures such as the gate fidelity, and can be used as a gate diagnostic tool. We apply our method to one- and two-qubit gates on two different superconducting systems available in the cloud, namely the QuTech Quantum Infinity and the IBM Quantum Experience. We discuss how cross-talk, leakage and non-Markovian errors affect the eigenvalue data.