Daniel Liang

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.

Daniel Liang

Given a dataset of input states, measurements, and probabilities, is it possible to efficiently predict the measurement probabilities associated with a quantum circuit? Recent work of Caro and Datta (2020) studied the problem of PAC learning quantum circuits in an information theoretic sense, leaving open questions of computational efficiency. In particular, one candidate class of circuits for which an efficient learner might have been possible was that of Clifford circuits, since the corresponding set of states generated by such circuits, called stabilizer states, are known to be efficiently PAC learnable (Rocchetto 2018). Here we provide a negative result, showing that proper learning of CNOT circuits is hard for classical learners unless $\textsf{RP} = \textsf{NP}$. As the classical analogue and subset of Clifford circuits, this naturally leads to a hardness result for Clifford circuits as well. Additionally, we show that if $\textsf{RP} = \textsf{NP}$ then there would exist efficient proper learning algorithms for CNOT and Clifford circuits. By similar arguments, we also find that an efficient proper quantum learner for such circuits exists if and only if $\textsf{NP} \subseteq \textsf{RQP}$.

Sabee Grewal, Daniel Liang

We study the problem of efficiently learning an unknown $n$-qubit unitary channel in diamond distance given query access. We present a general framework showing that if Pauli operators remain low-complexity under conjugation by a unitary, then the unitary can be learned efficiently. This framework yields polynomial-time algorithms for a wide range of circuit classes, including $O(\log \log n)$-depth circuits, quantum $O(\log n)$-juntas, near-Clifford circuits, the Clifford hierarchy, fermionic matchgate circuits, and certain compositions thereof. Our results unify and generalize prior work, and yield efficient learning algorithms for more expressive circuit classes than were previously known. Our framework is powered by new learning algorithms for unitaries whose Pauli spectrum is either supported on a small subgroup or is sparse. If the Pauli spectrum is supported on a subgroup of size $2^k$, we give an $\widetilde{O}(2^k/ε)$-query algorithm and a nearly matching $Ω(2^k/ε)$ lower bound. For $k = 2n$, we recover the optimal $O(4^n/ε)$-query algorithm of Haah, Kothari, O'Donnell, and Tang [FOCS '23]. If the Pauli spectrum is supported on $s$ Pauli operators, we give an $O(s^2/ε^2)$-query algorithm and an $Ω(s/ε)$ lower bound.

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.

Aravind Gollakota, Daniel Liang

We consider the problem of learning stabilizer states with noise in the Probably Approximately Correct (PAC) framework of Aaronson (2007) for learning quantum states. In the noiseless setting, an algorithm for this problem was recently given by Rocchetto (2018), but the noisy case was left open. Motivated by approaches to noise tolerance from classical learning theory, we introduce the Statistical Query (SQ) model for PAC-learning quantum states, and prove that algorithms in this model are indeed resilient to common forms of noise, including classification and depolarizing noise. We prove an exponential lower bound on learning stabilizer states in the SQ model. Even outside the SQ model, we prove that learning stabilizer states with noise is in general as hard as Learning Parity with Noise (LPN) using classical examples. Our results position the problem of learning stabilizer states as a natural quantum analogue of the classical problem of learning parities: easy in the noiseless setting, but seemingly intractable even with simple forms of noise.