We consider the resource bounded quantum circuit model with circuits restricted by thenumber of qubits they act upon and by their depth. Focusing on natural universal setsof gates which are familia...
We consider the resource bounded quantum circuit model with circuits restricted by the number of qubits they act upon and by their depth. Focusing on natural universal sets of gates which are familiar from classical circuit theory, several new lower bounds for constant depth quantum circuits are proved. The main result is that parity (and hence fanout) requires log depth quantum circuits, when the circuits are composed of single qubit and arbitrary size Toffoli gates, and when they use only constantly many ancill\ae. Under this constraint, this bound is close to optimal. In the case of a non-constant number a of ancill\ae\ and n input qubits, we give a tradeoff between a and the required depth, that results in a non-constant lower bound for fanout when a=n1−o(1). We also show that, regardless of the number of ancill\ae\, arbitrary arity Toffoli gates cannot be simulated exactly by a constant depth circuit family with gates of bounded arity.
This project focused on upper and lower bounds for quantum computability using constant depth quantum circuits, and on the complexity theory of such circuits. It established significant differences between resource bounded quantum and classical computation models, particularly emphasizing new examples of where quantum circuits are more powerful than their classical counterparts. A second focus and outgrowth of this work was the creation of efficient quantum algorithms for specific problems and central combinatorial functions. Recent finding include upper bounds for the computational power of constant depth circuits composed of single qubit and CNOT gates, and lower bounds for the computational power of constant depth circuits with single qubit, CNOT and fanout gates. These results represent the power and the limits of small, possibly realizable quantum circuits. Also examined were bounds on computations with additional storage qubits (ancillae), and natural simple functions (realized by small depth circuits) which require ancillae for their computation.
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.
The computational power of 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 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 circuits are universal: any QAC circuit can be exactly simulated by a two-dimensional geometrically local QAC circuit. Therefore, we focus on the computational power of 2D-QAC circuits with grid width w. We prove that PARITY computation has a tight width--depth tradeoff of 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) for PARITY would imply PARITY∈/QAC0. We also establish Fourier tail and sensitivity bounds, together with tight width lower bounds and tight depth lower bounds in 1D-QAC circuits for MAJORITY and 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 circuit increases this variance by at most a factor depending only on its width, yielding a variance bound for every 2D-QAC0 state. The high local measurement variance of cat and Dicke states then gives the width lower bounds.