Markus Heinrich

Properties of quantum systems can be estimated using classical shadows, which implement measurements based on random ensembles of unitaries. Originally derived for global Clifford unitaries and products of single-qubit Clifford gates, practical implementations are limited to the latter scheme for moderate numbers of qubits. Beyond local gates, the accurate implementation of very short random circuits with two-local gates is still experimentally feasible and, therefore, interesting for implementing measurements in near-term applications. In this work, we derive closed-form analytical expressions for shadow estimation using brickwork circuits with two layers of parallel two-local Haar-random (or Clifford) unitaries. Besides the construction of the classical shadow, our results give rise to sample-complexity guarantees for estimating Pauli observables. We then compare the performance of shadow estimation with brickwork circuits to the established approach using local Clifford unitaries and find improved sample complexity in the estimation of observables supported on sufficiently many qubits.

Abstract Many quantum information protocols require the implementation of random unitaries. Because it takes exponential resources to produce Haar-random unitaries drawn from the full n-qubit group, one often resorts to t-designs. Unitary t-designs mimic the Haar-measure up to t-th moments. It is known that Clifford operations can implement at most 3-designs. In this work, we quantify the non-Clifford resources required to break this barrier. We find that it suffices to inject O(t^4log^2(t)log(1/e)) many non-Clifford gates into a polynomial-depth random Clifford circuit to obtain an e-approximate t-design. Strikingly, the number of non-Clifford gates required is independent of the system size -- asymptotically, the density of non-Clifford gates is allowed to tend to zero. We also derive novel bounds on the convergence time of random Clifford circuits to the t-th moment of the uniform distribution on the Clifford group. Our proofs exploit a recently developed variant of Schur-Weyl duality for the Clifford group, as well as bounds on restricted spectral gaps of averaging operators.

Abstract Stabiliser operations occupy a prominent role in the theory of fault-tolerant quantum computing. They are defined operationally: by the use of Clifford gates, Pauli measurements and classical control. Within the stabiliser formalism, these operations can be efficiently simulated on a classical computer, a result which is known as the Gottesman-Knill theorem. However, an additional supply of magic states is enough to promote them to a universal, fault-tolerant model for quantum computing. To quantify the needed resources in terms of magic states, a resource theory of magic has been developed during the last years. Stabiliser operations (SO) are considered free within this theory, however they are not the most general class of free operations. From an axiomatic point of view, these are the completely stabiliser-preserving (CSP) channels, defined as those that preserve the convex hull of stabiliser states. It has been an open problem to decide whether these two definitions lead to the same class of operations. In this work, we answer this question in the negative, by constructing an explicit counter-example. This indicates that recently proposed stabiliser-based simulation techniques of CSP maps might be strictly more powerful than Gottesman-Knill-like methods. The result is analogous to a well-known fact in entanglement theory, namely that there is a gap between the class of local operations and classical communication (LOCC) and the class of separable channels. Along the way, we develop a number of auxiliary techniques which allow us to better characterise the set of CSP channels.

Many quantum information protocols require the implementation of random unitaries. Because it takes exponential resources to produce Haar-random unitaries drawn from the full n-qubit group, one often resorts to t-designs. Unitary t-designs mimic Haar-randomness up to t-th moments. It is known that Clifford operations can implement at most unitary 3-designs. In this work, we quantify the non-Clifford resources required to break this barrier. Exploiting a recently developed variant of Schur-Weyl duality for the Clifford group, wefind that it suffices to inject $O(t^4*\log^2(t), \log(1/\varepsilon))$ non-Clifford gates into a polynomial depth random Clifford circuit to obtain an ε-approximate t-design. Strikingly, the number n of qubits does not enter – asymptotically, the density of non-Clifford gates is allowed to tend to zero. As an auxiliary result that might be of independent interest, we obtain explicit bounds on the convergence time of random Clifford circuits to the t-th moment of the uniform distribution on the Clifford group.