arXiv subjects

Fengning Ou

Publications and source records attributed to Fengning Ou.

6 recordsLinked to original sources

arXiv2026

Optimal T Counts under Sparsity: from QROM to State Preparation and Block Encoding

Many quantum algorithms require coherent access to classical data, often modeled by quantum read-only memory (QROM). We initiate the study of the $T$ count of sparse QROM, in which only $s$ of the $2^n$ addresses store nonzero data. We prove asymptotically optimal $T$-count bounds $\Theta(\sqrt{sm} + \sqrt{sn})$ with square-root dependence on the support size $s$ and message length $m$. Our upper bounds use a multilevel hashing scheme, while our lower bounds reduce sparse QROM to state preparation and use counting arguments for adaptive Clifford+$T$ circuits. The lower bounds thus hold even when mid-circuit measurements and classically controlled operations are allowed. As applications, we obtain matching $T$-count bounds $\Theta(\sqrt{sn} + \sqrt{s\log(1/\varepsilon)} + \log(1/\varepsilon))$ for $s$-sparse state preparation and $\Theta( \sqrt{2^n sn} + \sqrt{2^n s\log(s/\varepsilon_{\mathrm{BE}})} + \log(s/\varepsilon_{\mathrm{BE}}))$ for block encoding of $s$-sparse matrices, where $\varepsilon$ and $\varepsilon_{\mathrm{BE}}$ are the precision of state preparation and block encoding, respectively.

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arXiv2026

Dimension-Free Approximate Tensorization of Quantum Hypercontractivity for Qudit Depolarizing Semigroups

We prove approximate tensorization for hypercontractivity and logarithmic-Sobolev constants for a class of primitive reversible quantum Markov semigroups satisfying the positive off-diagonal scaling (PODS) condition. This class includes qubit examples and generalized depolarizing semigroups with respect to full-rank states in arbitrary finite dimensions. For any such semigroup $(\Phi_t)_{t\ge 0}$ and every tensor power $n$, we show that the log-Sobolev constant of the product semigroup $\Phi_t^{\otimes n}$ is at least $2/(3\ln 2)\approx 0.96$ times the log-Sobolev constant of the single-site semigroup $\Phi_t$, independently of $n$ and the local dimension $d$. The proof first establishes an exact tensorization of the $(q,2)$-hypercontractive inequality for integer $q$, in particular $q=3$, and then extends the estimate to all real $q>2$ by complex interpolation; the standard implication from hypercontractivity to logarithmic-Sobolev inequalities yields the stated almost tensorization result. In the qubit case, we further prove exact tensorization for primitive reversible PODS semigroups and obtain sharp $(q,2)$-hypercontractivity estimates for generalized qubit depolarizing channels.

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arXiv2026

On the Computational Complexity of Geometrically Local QAC0 circuits

The computational complexity of $\mathsf{QAC}^0$, which are constant-depth, polynomial-size quantum circuit families consisting of arbitrary single-qubit unitaries and $n$-qubit generalized Toffoli gates, has gained tremendous focus recently. In this work, we initiate the study of the computational complexity of geometrically local $\mathsf{QAC}^0$ circuits, where all the generalized Toffoli gates act on nearest neighbor qubits. We show that any $\mathsf{QAC}^0$ circuit can be exactly simulated by a two-dimensional geometrically local $\mathsf{QAC}^0$ circuit, i.e., a $\mathsf{2D\text{-}QAC}^{0}$ circuit, with a quadratic size blow-up. This implies that $\mathsf{QAC}^0 = \mathsf{2D\text{-}QAC}^{0}$. We further show that if there existed a $\mathsf{QAC}^0$ circuit that computes Parity with a bounded constant error, then for any $\varepsilon > 0$, there would exist a $\mathsf{2D\text{-}QAC}^{0}$ circuit that exactly computes Parity, with a very "thin" width $n^\varepsilon$. We further study the computational power of $\mathsf{1D\text{-}QAC}^{0} $ circuits, i.e., one-dimensional $\mathsf{QAC}^0$ circuits, which are the "thinnest" $\mathsf{2D\text{-}QAC}^{0}$ circuits. We prove a nearly logarithmic depth lower bound on $\mathsf{1D\text{-}QAC}^{0} $ circuits to compute the Parity function, even if allowing an unlimited number of ancilla. Furthermore, if the inputs are encoded in contiguous qubits, we prove that it requires a nearly linear depth $\mathsf{1D\text{-}QAC}^{0} $ circuit to compute the Parity function. This lower bound is almost tight. The results are proved via the combination of the restriction argument and the light-cone argument. These results may provide a new angle for studying the computational power of $\mathsf{QAC}^0$ circuits and for resolving the long-standing open problem of whether Parity is in $\mathsf{QAC}^0$.

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arXiv2025

Linear-Size QAC0 Channels: Learning, Testing and Hardness

Shallow quantum circuits have attracted increasing attention in recent years, due to the fact that current noisy quantum hardware can only perform faithful quantum computation for a short amount of time. The constant-depth quantum circuits $\mathbf{QAC}^0$, a quantum counterpart of $\mathbf{AC}^0$ circuits, are the polynomial-size and constant-depth quantum circuits composed of only single-qubit unitaries and polynomial-size generalized Toffoli gates. The computational power of $\mathbf{QAC}^0$ has been extensively investigated in recent years. In this paper, we are concerned with $\mathbf{QLC}^0$ circuits, which are linear-size $\mathbf{QAC}^0$ circuits, a quantum counterpart of $\mathbf{LC}^0$. * We show that depth-$d$ $\mathbf{QAC}^0$ circuits working on $n$ input qubits and $a$ ancilla qubits have approximate degree at most $\tilde{O}((n+a)^{1-2^{-d}})$, improving the $\tilde{O}((n+a)^{1-3^{-d}})$ degree upper bound of previous works. Consequently, this directly implies that to compute the parity function, $\mathbf{QAC}^0$ circuits need at least $\tilde{O}(n^{1+2^{-d}})$ circuit size. * We present the first agnostic learning algorithm for $\mathbf{QLC}^0$ channels using subexponential running time and queries. Moreover, we also establish exponential lower bounds on the query complexity of learning $\mathbf{QAC}^0$ channels under both the spectral norm distance of the Choi matrix and the diamond norm distance. * We present a tolerant testing algorithm which determines whether an unknown quantum channel is a $\mathbf{QLC}^0$ channel. This tolerant testing algorithm is based on our agnostic learning algorithm. Our approach leverages low-degree approximations of $\mathbf{QAC}^0$ circuits and Pauli analysis as key technical tools. Collectively, these results advance our understanding of agnostic learning for shallow quantum circuits.

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arXiv2024

On the Computational Power of QAC0 with Barely Superlinear Ancillae

$\mathrm{QAC}^0$ is the family of constant-depth polynomial-size quantum circuits consisting of arbitrary single qubit unitaries and multi-qubit Toffoli gates. It was introduced by Moore [arXiv: 9903046] as a quantum counterpart of $\mathrm{AC}^0$, along with the conjecture that $\mathrm{QAC}^0$ circuits can not compute PARITY. In this work we make progress on this longstanding conjecture: we show that any depth-$d$ $\mathrm{QAC}^0$ circuit requires $n^{1+3^{-d}}$ ancillae to compute a function with approximate degree $\Theta(n)$, which includes PARITY, MAJORITY and $\mathrm{MOD}_k$. We further establish superlinear lower bounds on quantum state synthesis and quantum channel synthesis. This is the first superlinear lower bound on the super-linear sized $\mathrm{QAC}^0$. Regarding PARITY, we show that any further improvement on the size of ancillae to $n^{1+\exp(-o(d))}$ would imply that PARITY $\not\in$ QAC0. These lower bounds are derived by giving low-degree approximations to $\mathrm{QAC}^0$ circuits. We show that a depth-$d$ $\mathrm{QAC}^0$ circuit with $a$ ancillae, when applied to low-degree operators, has a degree $(n+a)^{1-3^{-d}}$ polynomial approximation in the spectral norm. This implies that the class $\mathrm{QLC}^0$, corresponding to linear size $\mathrm{QAC}^0$ circuits, has approximate degree $o(n)$. This is a quantum generalization of the result that $\mathrm{LC}^0$ circuits have approximate degree $o(n)$ by Bun, Robin, and Thaler [SODA 2019]. Our result also implies that $\mathrm{QLC}^0\neq\mathrm{NC}^1$.

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arXiv2023

Hypercontractivity for Quantum Erasure Channels via Variable Multipartite Log-Sobolev Inequality

We prove an almost optimal hypercontractive inequality for products of quantum erasure channels, generalizing the hypercontractivity for classical binary erasure channels. To our knowledge, this is the first tensorization-type hypercontractivity bound for quantum channels with no fixed states. The traditional inductive arguments for classical hypercontractivity cannot be generalized to the quantum setting due to the nature of the non-commutativity of matrices. To overcome the difficulty, we establish a novel quantum log-Sobolev inequality for Bernoulli entropy, which includes the classical log-Sobolev inequality and the quantum log-Sobolev inequality as one-partite cases. To our knowledge, its classical counterpart is also unknown prior to this work. We establish a connection between our quantum log-Sobolev inequality and the hypercontractivity bound for quantum erasure channels via a refined quantum Gross' lemma, extending the analogous connection between the quantum log-Sobolev inequality and the hypercontractivity for qubit unital channels. As an application, we prove an almost tight bound (up to a constant factor) on the classical communication complexity of two-party common randomness generation assisted with erased-noisy EPR states, generalizing the tight bound on the same task assisted with erased-noisy random strings due to Guruswami and Radhakrishnan.

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