Vishnu Iyer

Sabee Grewal, Vishnu Iyer, William Kretschmer et al.

We show that quantum states with "low stabilizer complexity" can be efficiently distinguished from Haar-random. Specifically, given an $n$-qubit pure state $|ψ\rangle$, we give an efficient algorithm that distinguishes whether $|ψ\rangle$ is (i) Haar-random or (ii) a state with stabilizer fidelity at least $\frac{1}{k}$ (i.e., has fidelity at least $\frac{1}{k}$ with some stabilizer state), promised that one of these is the case. With black-box access to $|ψ\rangle$, our algorithm uses $O\!\left( k^{12} \log(1/δ)\right)$ copies of $|ψ\rangle$ and $O\!\left(n k^{12} \log(1/δ)\right)$ time to succeed with probability at least $1-δ$, and, with access to a state preparation unitary for $|ψ\rangle$ (and its inverse), $O\!\left( k^{3} \log(1/δ)\right)$ queries and $O\!\left(n k^{3} \log(1/δ)\right)$ time suffice. As a corollary, we prove that $ω(\log(n))$ $T$-gates are necessary for any Clifford+$T$ circuit to prepare computationally pseudorandom quantum states, a first-of-its-kind lower bound.

Sabee Grewal, Vishnu Iyer, William Kretschmer et al.

We show that any pseudoentangled state ensemble with a gap of $t$ bits of entropy requires $Ω(t)$ non-Clifford gates to prepare. This bound is tight up to polylogarithmic factors if linear-time quantum-secure pseudorandom functions exist. Our result follows from a polynomial-time algorithm to estimate the entanglement entropy of a quantum state across any cut of qubits. When run on an $n$-qubit state that is stabilized by at least $2^{n-t}$ Pauli operators, our algorithm produces an estimate that is within an additive factor of $\frac{t}{2}$ bits of the true entanglement entropy.

Sabee Grewal, Vishnu Iyer, William Kretschmer et al.

Recent work has shown that $n$-qubit quantum states output by circuits with at most $t$ single-qubit non-Clifford gates can be learned to trace distance $ε$ using $\mathsf{poly}(n,2^t,1/ε)$ time and samples. All prior algorithms achieving this runtime use entangled measurements across two copies of the input state. In this work, we give a similarly efficient algorithm that learns the same class of states using only single-copy measurements.

Sabee Grewal, Vishnu Iyer, William Kretschmer et al.

We define a quantum learning task called agnostic tomography, where given copies of an arbitrary state $ρ$ and a class of quantum states $\mathcal{C}$, the goal is to output a succinct description of a state that approximates $ρ$ at least as well as any state in $\mathcal{C}$ (up to some small error $\varepsilon$). This task generalizes ordinary quantum tomography of states in $\mathcal{C}$ and is more challenging because the learning algorithm must be robust to perturbations of $ρ$. We give an efficient agnostic tomography algorithm for the class $\mathcal{C}$ of $n$-qubit stabilizer product states. Assuming $ρ$ has fidelity at least $τ$ with a stabilizer product state, the algorithm runs in time $n^{O(1 + \log(1/τ))} / \varepsilon^2$. This runtime is quasipolynomial in all parameters, and polynomial if $τ$ is a constant.

Sabee Grewal, Vishnu Iyer, William Kretschmer et al.

We give a pair of algorithms that efficiently learn a quantum state prepared by Clifford gates and $O(\log n)$ non-Clifford gates. Specifically, for an $n$-qubit state $|ψ\rangle$ prepared with at most $t$ non-Clifford gates, our algorithms use $\mathsf{poly}(n,2^t,1/\varepsilon)$ time and copies of $|ψ\rangle$ to learn $|ψ\rangle$ to trace distance at most $\varepsilon$. The first algorithm for this task is more efficient, but requires entangled measurements across two copies of $|ψ\rangle$. The second algorithm uses only single-copy measurements at the cost of polynomial factors in runtime and sample complexity. Our algorithms more generally learn any state with sufficiently large stabilizer dimension, where a quantum state has stabilizer dimension $k$ if it is stabilized by an abelian group of $2^k$ Pauli operators. We also develop an efficient property testing algorithm for stabilizer dimension, which may be of independent interest.