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Kumar used a switching lemma to prove exponential-size lower bounds for a circuit class GC^0 that not only contains AC^0 but can--with a single gate--compute functions that require exponential-size TC^0 circuits. His main result was that switching-lemma lower bounds for AC^0 lift to GC^0 with no loss in parameters, even though GC^0 requires exponential-size TC^0 circuits. Informally, GC^0 is AC^0 with unbounded-fan-in gates that behave arbitrarily inside a sufficiently small Hamming ball but must be constant outside it. We show an analogous result for GC^0[p] (GC^0 with MODp gates) and the polynomial method. Specifically, we show that polynomial-method lower bounds for AC^0[p] lift to GC^0[p] with no loss in parameters. As an application, we prove Majority requires depth-d GC^0[p] circuits of size $2^{\Omega(n^{1/2(d-1)})}$, matching the state-of-the-art lower bounds for AC^0[p]. We also show that E^NP requires exponential-size GCC^0 circuits (the union of GC^0[m] for all m), extending the result of Williams. It is striking that the switching lemma, polynomial method, and algorithmic method all generalize to GC^0-related classes, with the first two methods doing so without any loss. We also establish the strongest known unconditional separations between quantum and classical circuits: 1. There's an oracle relative to which BQP is not contained in the class of languages decidable by uniform families of size-$2^{n^{O(1)}}$ GC^0 circuits, generalizing Raz and Tal's relativized separation of BQP from the polynomial hierarchy. 2. There's a search problem that QNC^0 circuits can solve but average-case hard for exponential-size GC^0 circuits. 3. There's a search problem that QNC^0/qpoly circuits can solve but average-case hard for exponential-size GC^0[p] circuits. 4. There's an interactive problem that QNC^0 circuits can solve but exponential-size GC^0[p] circuits cannot., Comment: 47 pages

We define a quantum learning task called agnostic tomography, where given copies of an arbitrary state $\rho$ and a class of quantum states $\mathcal{C}$, the goal is to output a succinct description of a state that approximates $\rho$ 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 $\rho$. We give an efficient agnostic tomography algorithm for the class $\mathcal{C}$ of $n$-qubit stabilizer product states. Assuming $\rho$ has fidelity at least $\tau$ with a stabilizer product state, the algorithm runs in time $n^{O(1 + \log(1/\tau))} / \varepsilon^2$. This runtime is quasipolynomial in all parameters, and polynomial if $\tau$ is a constant., Comment: 20 pages. V2: minor corrections

We show that any pseudoentangled state ensemble with a gap of $t$ bits of entropy requires $\Omega(t)$ 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., Comment: 15 pages; v2: slight edits to concurrent work section

We define and study a variant of QMA (Quantum Merlin Arthur) in which Arthur can make multiple non-collapsing measurements to Merlin's witness state, in addition to ordinary collapsing measurements. By analogy to the class PDQP defined by Aaronson, Bouland, Fitzsimons, and Lee (2014), we call this class PDQMA. Our main result is that PDQMA = NEXP; this result builds on the MIP = NEXP Theorem and complements the result of Aaronson (2018) that PDQP/qpoly = ALL. While the result has little to do with quantum mechanics, we also show a more "quantum" result: namely, that QMA with the ability to inspect the entire history of a hidden variable is equal to NEXP, under mild assumptions on the hidden-variable theory. We also observe that a quantum computer, augmented with quantum advice and the ability to inspect the history of a hidden variable, can solve any decision problem in polynomial time., Comment: 15 pages

We introduce the entangled quantum polynomial hierarchy $\mathsf{QEPH}$ as the class of problems that are efficiently verifiable given alternating quantum proofs that may be entangled with each other. We prove $\mathsf{QEPH}$ collapses to its second level. In fact, we show that a polynomial number of alternations collapses to just two. As a consequence, $\mathsf{QEPH} = \mathsf{QRG(1)}$, the class of problems having one-turn quantum refereed games, which is known to be contained in $\mathsf{PSPACE}$. This is in contrast to the unentangled quantum polynomial hierarchy $\mathsf{QPH}$, which contains $\mathsf{QMA(2)}$. We also introduce a generalization of the quantum-classical polynomial hierarchy $\mathsf{QCPH}$ where the provers send probability distributions over strings (instead of strings) and denote it by $\mathsf{DistributionQCPH}$. Conceptually, this class is intermediate between $\mathsf{QCPH}$ and $\mathsf{QPH}$. We prove $\mathsf{DistributionQCPH} = \mathsf{QCPH}$, suggesting that only quantum superposition (not classical probability) increases the computational power of these hierarchies. To prove this equality, we generalize a game-theoretic result of Lipton and Young (1994) which says that the provers can send distributions that are uniform over a polynomial-size support. We also prove the analogous result for the polynomial hierarchy, i.e., $\mathsf{DistributionPH} = \mathsf{PH}$. These results also rule out certain approaches for showing $\mathsf{QPH}$ collapses. Finally, we show that $\mathsf{PH}$ and $\mathsf{QCPH}$ are contained in $\mathsf{QPH}$, resolving an open question of Gharibian et al. (2022)., Comment: 24 pages, 1 figure, 1 table

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 $\epsilon$ using $\mathsf{poly}(n,2^t,1/\epsilon)$ 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., Comment: This work has been merged into arXiv:2305.13409

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 $|\psi\rangle$ prepared with at most $t$ non-Clifford gates, our algorithms use $\mathsf{poly}(n,2^t,1/\varepsilon)$ time and copies of $|\psi\rangle$ to learn $|\psi\rangle$ to trace distance at most $\varepsilon$. The first algorithm for this task is more efficient, but requires entangled measurements across two copies of $|\psi\rangle$. 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., Comment: 54 pages. Merged v3 with arXiv:2308.07175. This version now subsumes arXiv:2308.07175

We study the complexity of learning quantum states in various models with respect to the stabilizer formalism and obtain the following results: - We prove that $\Omega(n)$ $T$-gates are necessary for any Clifford+$T$ circuit to prepare computationally pseudorandom quantum states, an exponential improvement over the previously known bound. This bound is asymptotically tight if linear-time quantum-secure pseudorandom functions exist. - Given an $n$-qubit pure quantum state $|\psi\rangle$ that has fidelity at least $\tau$ with some stabilizer state, we give an algorithm that outputs a succinct description of a stabilizer state that witnesses fidelity at least $\tau - \varepsilon$. The algorithm uses $O(n/(\varepsilon^2\tau^4))$ samples and $\exp\left(O(n/\tau^4)\right) / \varepsilon^2$ time. In the regime of $\tau$ constant, this algorithm estimates stabilizer fidelity substantially faster than the na\"ive $\exp(O(n^2))$-time brute-force algorithm over all stabilizer states. - In the special case of $\tau > \cos^2(\pi/8)$, we show that a modification of the above algorithm runs in polynomial time. - We exhibit a tolerant property testing algorithm for stabilizer states. The underlying algorithmic primitive in all of our results is Bell difference sampling. To prove our results, we establish and/or strengthen connections between Bell difference sampling, symplectic Fourier analysis, and graph theory., Comment: 41 pages, 2 figures. v3: changed presentation of tolerant testing algorithm and other minor edits

We show that quantum states with "low stabilizer complexity" can be efficiently distinguished from Haar-random. Specifically, given an $n$-qubit pure state $|\psi\rangle$, we give an efficient algorithm that distinguishes whether $|\psi\rangle$ 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 $|\psi\rangle$, our algorithm uses $O\!\left( k^{12} \log(1/\delta)\right)$ copies of $|\psi\rangle$ and $O\!\left(n k^{12} \log(1/\delta)\right)$ time to succeed with probability at least $1-\delta$, and, with access to a state preparation unitary for $|\psi\rangle$ (and its inverse), $O\!\left( k^{3} \log(1/\delta)\right)$ queries and $O\!\left(n k^{3} \log(1/\delta)\right)$ time suffice. As a corollary, we prove that $\omega(\log(n))$ $T$-gates are necessary for any Clifford+$T$ circuit to prepare computationally pseudorandom quantum states, a first-of-its-kind lower bound., Comment: 20 pages

We give an efficient algorithm that learns a non-interacting fermion state, given copies of the state. For a system of $n$ non-interacting fermions and $m$ modes, we show that $O(m^3 n^2 \log(1/\delta) / \epsilon^4)$ copies of the input state and $O(m^4 n^2 \log(1/\delta)/ \epsilon^4)$ time are sufficient to learn the state to trace distance at most $\epsilon$ with probability at least $1 - \delta$. Our algorithm empirically estimates one-mode correlations in $O(m)$ different measurement bases and uses them to reconstruct a succinct description of the entire state efficiently., Comment: 18 pages, 1 figure. We strengthen our results by learning the entire state, rather than the distribution, which is accomplished by a more careful error analysis and a slight modification to our algorithm. We also correct an error in the previous version (our analysis assumed the m*m matrix output by our algorithm was rank-n, but it was full-rank). We thank Andrew Zhao for identifying this error

The classical shadows protocol, recently introduced by Huang, Kueng, and Preskill [Nat. Phys. 16, 1050 (2020)], is a quantum-classical protocol to estimate properties of an unknown quantum state. Unlike full quantum state tomography, the protocol can be implemented on near-term quantum hardware and requires few quantum measurements to make many predictions with a high success probability. In this paper, we study the effects of noise on the classical shadows protocol. In particular, we consider the scenario in which the quantum circuits involved in the protocol are subject to various known noise channels and derive an analytical upper bound for the sample complexity in terms of a shadow seminorm for both local and global noise. Additionally, by modifying the classical post-processing step of the noiseless protocol, we define a new estimator that remains unbiased in the presence of noise. As applications, we show that our results can be used to prove rigorous sample complexity upper bounds in the cases of depolarizing noise and amplitude damping., Comment: 57 pages

We study the question of what is computable by Turing machines equipped with time travel into the past; i.e., with Deutschian closed timelike curves (CTCs) having no bound on their width or length. An alternative viewpoint is that we study the complexity of finding approximate fixed points of computable Markov chains and quantum channels of countably infinite dimension. Our main result is that the complexity of these problems is precisely $\Delta_2$, the class of languages Turing-reducible to the Halting problem. Establishing this as an upper bound for qubit-carrying CTCs requires recently developed results in the theory of quantum Markov maps., Comment: 25 pages; v1 contained an erroneous proof of the main theorem (Theorem 10). A correction is given in Theorem 3.3