In quantum theory, general measurements are described by Positive Operator-Valued Measures (POVMs). We show that for anyd-dimensional system, applying depolarizing noise with constant visibility rendersanyPOVM simulable by a randomized implementation of projective measurements— without auxiliary systems. This constrains the asymptotic advantage of POVMs over projective measurements in tasks such as state discrimination, shadow tomography, and quantum metrology. We further improve known bounds on the visibility range for which noisy pure two-qudit states admit local hidden-variable models under general measurements. As a byproduct, we obtain asymptotically tight bounds on the critical visibility for joint measurability of all POVMs. Technically, we leverage recent results in POVM simulation, the Kadison–Singer problem, and introduce a “dimension-deficient” Naimark theorem. Additionally, we demonstrate that general2N-qubit circuits can be simulated via randomization over circuits onN +1qubits, with constant overhead, suggesting new avenues for circuit knitting in near-term quantum architectures.

In this work, we present an in-depth study of average-case quantum distances introduced in Maciejewski et al. (2022). The average-case distances approximate, up to the relative error, the average Total-Variation (TV) distance between measurement outputs of two quantum processes, in which quantum objects of interest (states, measurements, or channels) are intertwined with random quantum circuits. Contrary to conventional distances, such as trace distance or diamond norm, they quantifyaverage-casestatistical distinguishability via random quantum circuits. We prove that once a family of random circuits forms an$\delta $-approximate 4-design, with$\delta =o(d^{-8})$, then the average-case distances can be approximated by simple explicit functions that can be expressed via simple degree two polynomials in objects of interest. For systems of moderate dimension, they can be easily explicitly computed – no optimization is needed as opposed to diamond norm distance between channels or operational distance between measurements. We prove that those functions, which we call quantum average-case distances, have a plethora of desirable properties, such as subadditivity w.r.t. tensor products, joint convexity, and (restricted) data-processing inequalities. Notably, all distances utilize the Hilbert-Schmidt (HS) norm, which provides this norm with a new operational interpretation. We also provide upper bounds on the maximal ratio between worst-case and average-case distances, and for each of them, we provide an example that saturates the bound. Specifically, we show that for each dimension$d$this ratio is at most$d^{\frac {1}{2}}, d, d^{\frac {3}{2}}$for states, measurements, and channels, respectively. To support the practical usefulness of our findings, we study multiple examples in which average-case quantum distances can be calculated analytically.

Epsilon-nets and approximate unitary$t$-designs are natural notions that capture properties of unitary operations relevant for numerous applications in quantum information and quantum computing. In this work we study quantitative connections between these two notions. Specifically, we prove that, for$d$dimensional Hilbert space, unitaries constituting$\delta $-approximate$t$-expanders form$\epsilon $-nets for$t\simeq \frac {d^{5/2}}{ \epsilon }$and$\delta \simeq \left ({\frac { \epsilon ^{3/2}}{d}}\right)^{d^{2}}$. We also show that for arbitrary$t$,$\epsilon $-nets can be used to construct$\delta $-approximate unitary$t$-designs for$\delta \simeq \epsilon t$, where the notion of approximation is based on the diamond norm. Finally, we prove that the degree of an exact unitary$t$design necessary to obtain an$\epsilon $-net must grow at least as fast as$\frac {1}{ \epsilon }$(for fixed dimension) and not slower than$d^{2}$(for fixed$\epsilon $). This shows near optimality of our result connecting$t$-designs and$\epsilon $-nets. We apply our findings in the context of quantum computing. First, we show that that approximate t-designs can be generated by shallow random circuits formed from a set of universal two-qudit gates in the parallel and sequential local architectures considered in (Brandão et al., 2016). Importantly, our gate sets need not to be symmetric (i.e., contains gates together with their inverses) or consist of gates with algebraic entries. Second, we consider compilation of quantum gates and prove a non-constructive Solovay-Kitaev theorem for general universal gate sets. Our main technical contribution is a new construction of efficient polynomial approximations to the Dirac delta in the space of quantum channels, which can be of independent interest.