We study the quantum complexity class QNC0fof quantum operations implement able exactly by constant-depth polynomial-size quantum circuits with unbounded fan-out gates. Our main result is that the quantum OR operation is in QNC0f, which is an affirmative answer to the question of Hoyer and Spalek. In sharp contrast to the strict hierarchy of the classical complexity classes: NC0⊊ AC0⊊ TC0, our result with Hoyer and Spalek's one implies the collapse of the hierarchy of the corresponding quantum ones: QNC0f= QAC0f= QTC0f. Then, we show that there exists a constant-depth sub quadratic-size quantum circuit for the quantum threshold operation. This allows us to obtain a better bound on the size difference between the QNC0fand QTC0fcircuits for implementing the same quantum operation. Lastly, we show that, if the quantum Fourier transform modulo a prime is in QNC0f, there exists a polynomial-time exact classical algorithm for a discrete logarithm problem using a QNC0foracle. This implies that, under a plausible assumption, there exists a classically hard problem that is solvable exactly by a QNC0fcircuit with gates for the quantum Fourier transform.
We study the quantum complexity class QNC0fof quantum operations implement able exactly by constant-depth polynomial-size quantum circuits with unbounded fan-out gates. Our main result is that the quantum OR operation is in QNC0f, which is an affirmative answer to the question of Hoyer and Spalek. In sharp contrast to the strict hierarchy of the classical complexity classes: NC0⊊ AC0⊊ TC0, our result with Hoyer and Spalek's one implies the collapse of the hierarchy of the corresponding quantum ones: QNC0f= QAC0f= QTC0f. Then, we show that there exists a constant-depth sub quadratic-size quantum circuit for the quantum threshold operation. This allows us to obtain a better bound on the size difference between the QNC0fand QTC0fcircuits for implementing the same quantum operation. Lastly, we show that, if the quantum Fourier transform modulo a prime is in QNC0f, there exists a polynomial-time exact classical algorithm for a discrete logarithm problem using a QNC0foracle. This implies that, under a plausible assumption, there exists a classically hard problem that is solvable exactly by a QNC0fcircuit with gates for the quantum Fourier transform.
We study the quantum complexity class QNC^0_f of quantum operations implementable exactly by constant-depth polynomial-size quantum circuits with unbounded fan-out gates (called QNC^0_f circuits). Our main result is that the quantum OR operation is in QNC^0_f, which is an affirmative answer to the question of Hoyer and Spalek. In sharp contrast to the strict hierarchy of the classical complexity classes: NC^0 \subsetneq AC^0 \subsetneq TC^0, our result with Hoyer and Spalek's one implies the collapse of the hierarchy of the corresponding quantum ones: QNC^0_f = QAC^0_f = QTC^0_f. Then, we show that there exists a constant-depth subquadratic-size quantum circuit for the quantum threshold operation. This implies the size difference between the QNC^0_f and QTC^0_f circuits for implementing the same quantum operation. Lastly, we show that, if the quantum Fourier transform modulo a prime is in QNC^0_f, there exists a polynomial-time exact classical algorithm for a discrete logarithm problem using a QNC^0_f oracle. This implies that, under a plausible assumption, there exists a classically hard problem that is solvable exactly by a QNC^0_f circuit with gates for the quantum Fourier transform.
Abstract. We demonstrate that the unbounded fan-out gate is very powerful. Constant-depth polynomial-size quantum circuits with bounded fan-in and unbounded fan-out over a fixed basis (denoted by QNC 0 f) can approximate with polynomially small error the following gates: parity, mod[q], And, Or, majority, threshold[t], exact[q], and counting. Classically, we need logarithmic depth even if we can use unbounded fan-in gates. If we allow arbitrary one-qubit gates instead of a fixed basis, then these circuits can also be made exact in log-star depth. Sorting, arithmetical operations, phase estimation, and the quantum Fourier transform can also be approximated in constant depth. 1
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.
We demonstrate that the unbounded fan-out gate is very powerful. Constant-depth polynomial-size quantum circuits with bounded fan-in and unbounded fan-out over a fixed basis (denoted by QNCf0) can approximate with polynomially small error the following gates: parity, mod[q], And, Or, majority, threshold[t], exact[t], and Counting. Classically, we need logarithmic depth even if we can use unbounded fan-in gates. If we allow arbitrary one-qubit gates instead of a fixed basis, then these circuits can also be made exact in log-star depth. Sorting, arithmetic operations, phase estimation, and the quantum Fourier transform with arbitrary moduli can also be approximated in constant depth.
We demonstrate that the unbounded fan-out gate is very powerful.Constantdepth polynomial-size quantum circuits with bounded fan-in and unbounded fan-out over a fixed basis (denoted by QNC 0 f ) can approximate with polynomially small error the following gates: parity, mod[q], And, Or, majority, threshold[t], exact[t], and Counting.Classically, we need logarithmic depth even if we can use unbounded fan-in gates.If we allow arbitrary one-qubit gates instead of a fixed basis, then these circuits can also be made exact in log-star depth.Sorting, arithmetic operations, phase estimation, and the quantum Fourier transform with arbitrary moduli can also be approximated in constant depth.
We propose definitions of QAC^0, the quantum analog of the classical class AC^0 of constant-depth circuits with AND and OR gates of arbitrary fan-in, and QACC[q], the analog of the class ACC[q] where Mod_q gates are also allowed. We prove that parity or fanout allows us to construct quantum MOD_q gates in constant depth for any q, so QACC[2] = QACC. More generally, we show that for any q,p > 1, MOD_q is equivalent to MOD_p (up to constant depth and polynomial size). This implies that QAC^0 with unbounded fanout gates, denoted QACwf^0, is the same as QACC[q] and QACC for all q. Since \ACC[p] \ne ACC[q] whenever p and q are distinct primes, QACC[q] is strictly more powerful than its classical counterpart, as is QAC^0 when fanout is allowed. This adds to the growing list of quantum complexity classes which are provably more powerful than their classical counterparts. We also develop techniques for proving upper bounds for QACC in terms of related language classes. We define classes of languages closely related to QACC[2] and show that restricted versions of them can be simulated by polynomial-size circuits. With further restrictions, language classes related to QACC[2] operators can be simulated by classical threshold circuits of polynomial size and constant depth.