David Gross

Gaussian states are widely regarded as the most important class of continuous-variable (CV) quantum states, as they naturally arise in physical systems and play a key role in quantum technologies. This motivates a fundamental question: given copies of an unknown CV state, how can we efficiently test whether it is Gaussian? We address this problem from the perspective of representation theory and quantum learning theory, characterizing the sample complexity of Gaussianity testing as a function of the number of modes. For pure states, we prove that just a constant number of copies is sufficient to decide whether the state is exactly Gaussian. We then extend this to the tolerant setting, showing that a polynomial number of copies suffices to distinguish states that are close to Gaussian from those that are far. In contrast, we establish that testing Gaussianity of general mixed states necessarily requires exponentially many copies, thereby identifying a fundamental limitation in testing CV systems. Our approach relies on rotation-invariant symmetries of Gaussian states together with the recently introduced toolbox of CV trace-distance bounds.

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.

In low-rank matrix recovery, one aims to reconstruct a low-rank matrix from a minimal number of linear measurements. Within the paradigm of compressed sensing, this is made computationally efficient by minimizing the nuclear norm as a convex surrogate for rank. In this work, we identify an improved regularizer based on the so-called diamond norm, a concept imported from quantum information theory. We show that -for a class of matrices saturating a certain norm inequality- the descent cone of the diamond norm is contained in that of the nuclear norm. This suggests superior reconstruction properties for these matrices. We explicitly characterize this set of matrices, which also contains quantum channels. Moreover, we demonstrate numerically that the diamond norm indeed outperforms the nuclear norm in a number of relevant applications: These include not only the task of quantum process tomography but also signal analysis tasks such as blind matrix deconvolution or the retrieval of certain unitary basis changes. The diamond norm is defined for matrices that can be interpreted as order-4 tensors and it turns out that the above condition depends crucially on that tensorial structure. In this sense, this work touches on an aspect of the notoriously difficult tensor completion problem.