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arXiv:2609.34659

Depth-Optimal Quantum Compilation

Francisca Vasconcelos

quant-phcs.CCSubmitted Sep 28, 2026• Announced Sep 29, 2026• 0 citations
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Abstract

We achieve the first constant-depth circuit for arbitrary single-qubit gate synthesis. Unlike prior approaches, the construction is fully unitary and requires no pre-supplied catalyst. For any constant $δ>0$, it $\varepsilon$-approximates an arbitrary single-qubit gate using $O(\log^{1+δ}(1/\varepsilon))$ clean ancillae, Hadamard and $T$ single-qubit gates, $O(\log(1/\varepsilon))$-width generalized Toffoli gates, and sublogarithmic-width Fan-Out gates. We further eliminate Fan-Out entirely, showing that Hadamard, $T$, and generalized Toffoli gates alone suffice for constant-depth synthesis. When restricted to the standard bounded-width gate model, our construction has depth $O(\log\log(1/\varepsilon))$, and we prove a matching $Ω(\log\log(1/\varepsilon))$-depth lower bound. Overall, we establish that $Θ(\log\log(1/\varepsilon))$-depth is unavoidable with only bounded-width gates, yet allowing even logarithmic-width multi-qubit gates suffices to achieve constant-depth synthesis. These results also reveal new structure in shallow quantum circuit complexity. We give a depth-preserving real simulation of bounded-error decision computation, showing that every depth-$d$ QAC circuit can be simulated in depth $O(d)$ using only Hadamard, $X$, and generalized Toffoli gates. Thus arbitrary single-qubit rotations and complex amplitudes do not increase the bounded-error decision power of QAC, even at constant depth. In particular, this reduces the long-standing conjecture Parity$\notin$QAC$^0$ to proving a Parity lower bound against circuits consisting only of Hadamard, $X$, and generalized Toffoli gates. More generally, this real normal form exposes a direct correspondence between the standard shallow-depth quantum circuit hierarchy and a hierarchy of Forrelation circuits with restricted oracle families.

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All Unitaries Have Constant Depth Quantum Circuits

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Learning shallow quantum circuits with many-qubit gates

Francisca Vasconcelos, Hsin-Yuan Huang

We present the first computationally-efficient algorithm for average-case learning of shallow quantum circuits with many-qubit gates. Specifically, we provide a quasi-polynomial time and sample complexity algorithm for learning unknown QAC$^0$ circuits -- constant-depth circuits with arbitrary single-qubit gates and polynomially many $CZ$ gates of unbounded width -- with at most logarithmic ancilla, up to inverse-polynomially small error. Furthermore, we show that the learned unitary can be efficiently synthesized in poly-logarithmic depth. This work expands the family of efficiently learnable quantum circuits, notably since in finite-dimensional circuit geometries, QAC$^0$ circuits require polynomial depth to implement.

arXiv:2410.16693•Oct 22, 2024•1 citation
We achieve the first constant-depth circuit for arbitrary single-qubit gate synthesis. Unlike prior approaches, the construction is fully unitary and requires no pre-supplied catalyst. For any constant δ>0δ>0δ>0, it ε\varepsilonε-approximates an arbitrary single-qubit gate using O(log⁡1+δ(1/ε))O(\log^{1+δ}(1/\varepsilon))O(log1+δ(1/ε)) clean ancillae, Hadamard and TTT single-qubit gates, O(log⁡(1/ε))O(\log(1/\varepsilon))O(log(1/ε))-width generalized Toffoli gates, and sublogarithmic-width Fan-Out gates. We further eliminate Fan-Out entirely, showing that Hadamard, TTT, and generalized Toffoli gates alone suffice for constant-depth synthesis. When restricted to the standard bounded-width gate model, our construction has depth O(log⁡log⁡(1/ε))O(\log\log(1/\varepsilon))O(loglog(1/ε)), and we prove a matching Ω(log⁡log⁡(1/ε))Ω(\log\log(1/\varepsilon))Ω(loglog(1/ε))-depth lower bound. Overall, we establish that Θ(log⁡log⁡(1/ε))Θ(\log\log(1/\varepsilon))Θ(loglog(1/ε))-depth is unavoidable with only bounded-width gates, yet allowing even logarithmic-width multi-qubit gates suffices to achieve constant-depth synthesis.
These results also reveal new structure in shallow quantum circuit complexity. We give a depth-preserving real simulation of bounded-error decision computation, showing that every depth-ddd QAC circuit can be simulated in depth O(d)O(d)O(d) using only Hadamard, XXX, and generalized Toffoli gates. Thus arbitrary single-qubit rotations and complex amplitudes do not increase the bounded-error decision power of QAC, even at constant depth. In particular, this reduces the long-standing conjecture Parity∉\notin∈/QAC0^00 to proving a Parity lower bound against circuits consisting only of Hadamard, XXX, and generalized Toffoli gates. More generally, this real normal form exposes a direct correspondence between the standard shallow-depth quantum circuit hierarchy and a hierarchy of Forrelation circuits with restricted oracle families.
Classical circuits with unbounded fan-in can compute any Boolean function in constant depth when their size is unrestricted. We ask whether removing the restrictions on circuit size and ancillary qubits also allows quantum circuits built from arbitrary single-qubit gates and generalised Toffoli gates to implement every unitary in constant depth. We give exact constant-depth constructions for arbitrary permutations of computational basis states, diagonal unitaries and the preparation of arbitrary pure states. These connect quantum state preparation to reversible classical computation and the preparation of probability distributions. With fanout in the gate set, they use exponentially many gates and ancillary qubits and return all ancillary qubits to zero. Replacing fanout by an exact circuit over the original gate set preserves constant depth, although the size bounds can become doubly exponential. The implementation of arbitrary unitaries in constant depth remains open. We give equivalent formulations in terms of copying the vectors of a specified orthonormal basis, extracting their labels and implementing restricted families of unitaries. We also reduce arbitrary unitary implementation to that of traceless unitary involutions using one additional clean qubit. With adaptive measurements, gate teleportation gives depth proportional to the level of a gate in the Clifford hierarchy. Towards arbitrary unitary implementation in constant depth, we use port-based teleportation: for input dimension ddd and M≥d2−1M\geq d^2-1M≥d2−1 ports, we construct a unitary circuit of depth O(d)O(\sqrt d)O(d​), independent of MMM and including resource preparation and port selection, with entanglement fidelity at least (1−(d2−1)/(2M))2(1-(d^2-1)/(2M))^2(1−(d2−1)/(2M))2. Thus, at fixed ddd, the approximation can be made arbitrarily accurate without increasing depth. Whether the dependence on ddd can also be removed remains open.
Circuit depth is a central resource in complexity theory. While bounded-depth classical circuits admit well-understood hierarchy theorems, the internal structure of constant-depth quantum computation remains comparatively unexplored. We prove an explicit depth hierarchy theorem for QNC0\mathsf{QNC}^0QNC0. For each d≥12d\ge 12d≥12, we construct a family of two-round interactive problems on which no depth-(d−1)(d-1)(d−1) quantum circuit can achieve near-perfect success, regardless of gate set, circuit size, or ancillary qubits. In contrast, we prove that our construction admits realizations by simple bounded fan-in quantum circuits of depth larger than ddd by a small constant factor. Moreover, all bounded fan-in classical circuits of sublogarithmic depth (in the input size) fail to achieve perfect success on these tasks for every ddd, yielding a hierarchy of problems that show unconditional quantum advantage of QNC0\mathsf{QNC}^0QNC0 over NC0\mathsf{NC}^0NC0. A key obstacle is the scarcity of lower bound techniques for quantum circuits. To address this, we develop methods to analyze how depth affects a circuit's ability to realize nonlocal correlations amongst its output qubits in a fine-grained manner. Our approach exploits the correspondence between constraint systems and nonlocal games, translating group-theoretic constructions into rigid operator-valued constraint systems and then into non-local games. In particular, we construct constraint systems whose unique faithful operator-valued solutions require every perfect strategy, and every near-perfect strategy to a fixed precision, to implement multi-controlled phase operations. This reduces to a nonlocal unitary-synthesis problem, yielding depth lower bounds for both shallow quantum and classical circuits. These results show that increasing depth strictly increases computational power within QNC0\mathsf{QNC}^0QNC0, establishing a genuinely quantum hierarchy.
The computational power of QAC\mathsf{QAC}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 QAC\mathsf{QAC}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 2D-QAC\mathsf{2D\text{-}QAC}2D-QAC circuits are universal: any QAC\mathsf{QAC}QAC circuit can be exactly simulated by a two-dimensional geometrically local QAC\mathsf{QAC}QAC circuit. Therefore, we focus on the computational power of 2D-QAC\mathsf{2D\text{-}QAC}2D-QAC circuits with grid width www. We prove that PARITY\mathsf{PARITY}PARITY computation has a tight width--depth tradeoff of w=nΘ(1/d)w=n^{Θ(1/d)}w=nΘ(1/d) in the constant-depth regime if the circuit allows arbitrary ancilla. Moreover, for polynomial-size circuits, a width lower bound of nω(1/d)n^{ω(1/d)}nω(1/d) for PARITY\mathsf{PARITY}PARITY would imply PARITY∉QAC0\mathsf{PARITY}\notin\mathsf{QAC}^0PARITY∈/QAC0. We also establish Fourier tail and sensitivity bounds, together with tight width lower bounds and tight depth lower bounds in 1D-QAC\mathsf{1D\text{-}QAC}1D-QAC circuits for MAJORITY\mathsf{MAJORITY}MAJORITY and MOD\mathsf{MOD}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 2D-QAC0\mathsf{2D\text{-}QAC}^02D-QAC0 circuit increases this variance by at most a factor depending only on its width, yielding a variance bound for every 2D-QAC0\mathsf{2D\text{-}QAC}^02D-QAC0 state. The high local measurement variance of cat and Dicke states then gives the width lower bounds.
Mid-circuit measurements with feedforward reduce the optimal worst-case quantum depth for exact nnn-qubit unitary synthesis from Θ(n)Θ(n)Θ(n) to O(1)O(1)O(1), using arbitrary one- and two-qubit gates and unrestricted ancillary space. Measurements and resets without feedforward remain subject to the circuit causal-cone bound. Our deterministic construction uses single-qubit measurements, O(1)O(1)O(1) rounds of parity feedforward, and O(n4nlog⁡(n+2))O(n4^n\log(n+2))O(n4nlog(n+2)) total qubits, with all ancillas starting and ending in ∣0⟩|0\rangle∣0⟩. Classical processing is not counted in quantum depth. We teleport only the matrix error of an approximation by Nehoran and Yuen. An exactly prepared ancillary state encodes this error, whose action is added back by coherent interference. At operator-norm error 2−n2^{-n}2−n, one step of amplitude amplification suffices. Coherent fanout circuits achieve the same constant-depth bound, while circuits using only one- and two-qubit gates attain depth Θ(n)Θ(n)Θ(n) with O(4nlog⁡2(n+2))O(4^n\log^2(n+2))O(4nlog2(n+2)) gates. Generic targets require Ω(4n/(h+1))Ω(4^n/(h+1))Ω(4n/(h+1)) qubits at depth hhh, even under unrestricted classical control. Our depth-width trade-offs meet this bound within a factor O(n2)O(n^2)O(n2) for 1≤h≤2n1\le h\le2^n1≤h≤2n.
It is well-known that every nnn-qubit unitary can be implemented by a 2O(n)2^{O(n)}2O(n)-depth quantum circuit using single- and two-qubit gates. It has been open whether exponential depth is *necessary* for general unitaries, even when allowing an unlimited number of ancilla qubits. Here we show, perhaps surprisingly, that all unitaries can be implemented exactly by a circuit of one- and two-qubit gates of depth poly(n)\mathsf{poly}(n)poly(n) with 2O(n)2^{O(n)}2O(n) ancilla qubits. In other words, every nnn-qubit unitary can be parallelized to polynomial depth. In fact, our depth bound is *linear* in nnn, which is the best possible, and an exponential improvement on the previous best bound of 2n/22^{n/2}2n/2 due to Rosenthal [TQC 2022, Quantum 2026]. Moreover, if we allow unbounded fan-out gates, these circuits can be further reduced to *constant* depth. Our construction takes advantage of a novel relationship connecting the unitary synthesis problem of Aaronson and Kuperberg to locally-decodable codes and private information retrieval from complexity theory and cryptography, and has a natural interpretation in bosonic quantum computation.
We present the first computationally-efficient algorithm for average-case learning of shallow quantum circuits with many-qubit gates. Specifically, we provide a quasi-polynomial time and sample complexity algorithm for learning unknown QAC0^00 circuits -- constant-depth circuits with arbitrary single-qubit gates and polynomially many CZCZCZ gates of unbounded width -- with at most logarithmic ancilla, up to inverse-polynomially small error. Furthermore, we show that the learned unitary can be efficiently synthesized in poly-logarithmic depth. This work expands the family of efficiently learnable quantum circuits, notably since in finite-dimensional circuit geometries, QAC0^00 circuits require polynomial depth to implement.