It is well-known that every $n$-qubit unitary can be implemented by a $2^{O(n)}$-depth quantum circuit using single- and two-qubit gates. It has been open whether exponential depth is \emph{necessary} for general unitaries, even when allowing for unlimited number of ancilla qubits. We show, perhaps surprisingly, that all unitaries can be approximated to operator norm $ε$ by a circuit of one- and two-qubit gates of depth $\poly(n,\log 1/ε)$ with $2^{O(n)}$ ancilla qubits. In other words, every $n$-qubit unitary can be parallelized to polynomial depth. Moreover, if we allow unbounded fan-out gates, these circuits can be reduced further to \emph{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.
It is well-known that every n-qubit unitary can be implemented by a 2O(n)-depth quantum circuit using single- and two-qubit gates. It has been open whether exponential depth is \emph{necessary} for general unitaries, even when allowing for unlimited number of ancilla qubits. We show, perhaps surprisingly, that all unitaries can be approximated to operator norm ε by a circuit of one- and two-qubit gates of depth \poly(n,log1/ε) with 2O(n) ancilla qubits. In other words, every n-qubit unitary can be parallelized to polynomial depth. Moreover, if we allow unbounded fan-out gates, these circuits can be reduced further to \emph{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.
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 d and M≥d2−1 ports, we construct a unitary circuit of depth O(d), independent of M and including resource preparation and port selection, with entanglement fidelity at least (1−(d2−1)/(2M))2. Thus, at fixed d, the approximation can be made arbitrarily accurate without increasing depth. Whether the dependence on d can also be removed remains open.
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 ε-approximates an arbitrary single-qubit gate using O(log1+δ(1/ε)) clean ancillae, Hadamard and T single-qubit gates, O(log(1/ε))-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(loglog(1/ε)), and we prove a matching Ω(loglog(1/ε))-depth lower bound. Overall, we establish that 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-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∈/QAC0 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.
Mid-circuit measurements with feedforward reduce the optimal worst-case quantum depth for exact n-qubit unitary synthesis from Θ(n) to 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) rounds of parity feedforward, and O(n4nlog(n+2)) total qubits, with all ancillas starting and ending in ∣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−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) with O(4nlog2(n+2)) gates. Generic targets require Ω(4n/(h+1)) qubits at depth h, even under unrestricted classical control. Our depth-width trade-offs meet this bound within a factor O(n2) for 1≤h≤2n.
We prove that any n -qubit unitary can be implemented (i) approximately in time O ~ ( 2 n / 2 ) with query access to an appropriate classical oracle, and also (ii) exactly by a circuit of depth O ~ ( 2 n / 2 ) with one- and two-qubit gates and 2 O ( n ) ancillae. The proofs involve similar reductions to Grover search. The proof of (ii) also involves a linear-depth construction of arbitrary quantum states using one- and two-qubit gates (in fact, this can be improved to constant depth with the addition of fanout and generalized Toffoli gates) which may be of independent interest. We also prove a matching Ω ( 2 n / 2 ) lower bound for (i) and (ii) for a certain class of implementations.
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. For each d≥12, we construct a family of two-round interactive problems on which no depth-(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 d 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 d, yielding a hierarchy of problems that show unconditional quantum advantage of QNC0 over NC0.
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, establishing a genuinely quantum hierarchy.
Unitary synthesis underlies universal quantum computation by translating unitary transformations into circuits of elementary gates. We show that for any budget m≥0 of ancilla, every n-qubit unitary can be exactly implemented using a quantum circuit consisting of O(4n) CNOT and single-qubit gates in O(n+4n/(n+m)) layers attaining the asymptotically optimal depth--ancilla tradeoff while simultaneously achieving optimal circuit size. The key idea is to encode computational basis states as wavefunctions sharing a Gaussian envelope, converting unitary action on amplitudes into coordinate transformations that admit shallow circuit implementations. The continuous encoding is then discretized, with approximation errors removed by exact residual correction. The depth-ancilla tradeoff follows by recursively applying the quantum Shannon decomposition, synthesizing its smaller unitary factors, and reusing ancillary qubits.