arXiv subjects

Fengning Ou

Publications and source records attributed to Fengning Ou.

8 recordsLinked to original sources

arXiv2026

unitaryQIP = unitaryPSPACE

Unitary synthesis complexity studies the computational cost of applying a target unitary to an unknown quantum state, extending complexity theory beyond its traditional focus on problems with classical inputs and outputs. Rosenthal and Yuen (ITCS 2022) introduced two complexity classes for unitary synthesis: $\mathsf{unitaryQIP}$ and $\mathsf{unitaryPSPACE}$ for the unitary families implementable via interactive proofs and in polynomial space, respectively. In this work, we prove the equality $\mathsf{unitaryQIP}=\mathsf{unitaryPSPACE}$ and thereby establish a unitary-complexity analogue of $\mathsf{QIP} = \mathsf{PSPACE}$ (Jain, Ji, Upadhyay, and Watrous, J. ACM 2011) and $\mathsf{stateQIP}=\mathsf{statePSPACE}$ (Rosenthal and Yuen, ITCS 2022; Metger and Yuen, FOCS 2023). To prove $\mathsf{unitaryQIP} \subseteq \mathsf{unitaryPSPACE}$, we use an acceptance rigidity lemma derived from the soundness condition to show that the polynomial-space prover constructed for a fixed input in state synthesis (Metger and Yuen, FOCS 2023) implements the target unitary on every input, leading to a polynomial-space simulation of the protocol. For the reverse inclusion, we build on OpenAI's recently announced solution to the unitary synthesis problem (Aaronson and Kuperberg, Theory Comput. 2007), and show that $\mathsf{unitaryPSPACE}$ families can be synthesized in polynomial time with a $\mathsf{PSPACE}$ oracle. We then implement the oracle queries using auxiliary states prepared via the $\mathsf{statePSPACE} \subseteq \mathsf{stateQIP}$ protocol (Rosenthal and Yuen, ITCS 2022). Beyond the equality, we show that our protocol for $\mathsf{unitaryPSPACE}$ can be parallelized to constant rounds when the verifier receives polynomially many copies of the input, and we also obtain a complexity-theoretic barrier to proving the existence of $\mathsf{EFI}$ pairs.

quant-ph↗

arXiv2026

We Query, Therefore We Compute: On Oracle Computation beyond the Machine, with an Application to Agents

Agentic systems use large language models (LLMs) to carry out concrete tasks. Prior work often borrows abstractions such as scheduling, caching or isolation piecemeal from operating systems, so the mechanisms it builds share little common ground, and the shared view of the two forms of agentic system, Workflows and Agents, is limited. We construct an abstract machine that provides both. We treat the LLM as an Oracle and extend a two-stack pushdown automaton with one instruction, which hands the Oracle a whole stack as its query and appends the answer to that same stack. The machine thus performs two computations, the Oracle's and a Turing-complete one that we call the Priestess. A stack that the program only appends to grows autoregressively, as an agent's context does. Two symmetry breakings, S in storage and T in transitions, make a Priestess program the operating system of the programs the Oracle runs, and produce the Agent and the Workflow as the two placements of a task's program. For internally autoregressive Oracles, the two computations synchronize at the end of every answer under certain conditions, and through that synchronization we model caching and analyse scheduling. No guarantee that holds for every Oracle can fix which content crosses between the two computations, but such a guarantee does fix the boundary itself. The construction V fits the machine to a von Neumann computer. To show that it is realizable, we propose ArchNights, an extended RISC-V ISA and a Linux-style operating system implementing the machine by design. ArchNights-SE runs on gem5 as a computer system, becomes an agentic system when it runs an LLM as the Oracle, and will be open source. Agentic systems can then be designed as computer systems are. With a foundation built and a unified view, future work can share invariants and bounds, each with its conditions.

cs.AI↗

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 $\mathrm{T}$ count of sparse QROM, in which only $s$ of the $2^n$ addresses store nonzero data. We prove that the optimal $\mathrm{T}$ count is $Θ\left(n+\min\left\{s,\sqrt{s\left(m+\log(2^{n+1}/s)\right)}\right\}\right)$. 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+$\mathrm{T}$ circuits. The lower bounds thus hold even when mid-circuit measurements and classically controlled operations are allowed. As applications, we obtain matching $\mathrm{T}$-count bounds $Θ\left(\min\left\{s,\sqrt{s\log(2^{n+1}/s)}\right\} +\sqrt{s\log(1/\varepsilon)}+\log(1/\varepsilon)\right)$ for $s$-sparse state preparation and $Θ\left(\sqrt{2^n s\left(n+\log(1/\varepsilon_{\rm BE})\right)} +\log(1/\varepsilon_{\rm BE})\right)$ for block encoding of $s$-sparse matrices, where $\varepsilon$ and $\varepsilon_{\rm BE}$ are the precision of state preparation and block encoding, respectively.

quant-ph↗

arXiv2026

On the Computational Power of Geometrically Local QAC circuits

The computational power of $\mathsf{QAC}$, which are constant-depth 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 $\mathsf{QAC}$ circuits, where the qubits are arranged in a high-dimensional grid, and all the generalized Toffoli gates act on contiguous qubits along one direction. We prove that $\mathsf{2D\text{-}QAC}$ circuits are universal: any $\mathsf{QAC}$ circuit can be exactly simulated by a two-dimensional geometrically local $\mathsf{QAC}$ circuit. Therefore, we focus on the computational power of $\mathsf{2D\text{-}QAC}$ circuits with grid width $w$. We prove that $\mathsf{PARITY}$ computation has a tight width--depth tradeoff of $w=n^{Θ(1/d)}$ in the constant-depth regime if the circuit allows arbitrary ancilla. Moreover, for polynomial-size circuits, a width lower bound of $n^{ω(1/d)}$ for $\mathsf{PARITY}$ would imply $\mathsf{PARITY}\notin\mathsf{QAC}^0$. We also establish Fourier tail and sensitivity bounds, together with tight width lower bounds and tight depth lower bounds in $\mathsf{1D\text{-}QAC}$ circuits for $\mathsf{MAJORITY}$ and $\mathsf{MOD}$ computation. Our main tool is \emph{local measurement variance}, the maximum variance of normalized sums of observables on separated regions. We use a hybrid argument to show that each layer of a $\mathsf{2D\text{-}QAC}^0$ circuit increases this variance by at most a factor depending only on its width, yielding a variance bound for every $\mathsf{2D\text{-}QAC}^0$ state. The high local measurement variance of cat and Dicke states then gives the width lower bounds.

quant-ph↗

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 $(Φ_t)_{t\ge 0}$ and every tensor power $n$, we show that the log-Sobolev constant of the product semigroup $Φ_t^{\otimes n}$ is at least $2/(3\ln 2)\approx 0.96$ times the log-Sobolev constant of the single-site semigroup $Φ_t$, 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.

quant-ph↗

arXiv2025

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 $Θ(n)$, 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 $\mathrm{QLC}^0\neq\mathrm{NC}^1$.

quant-ph↗

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.

quant-ph↗

arXiv2025

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.

quant-ph↗