We present a technique to derive depth lower bounds for quantum circuits. The technique is based on the observation that in circuits without ancillæ, only a few input states can set all the control qubits of a Toffoli gate to 1. This can be used to selectively remove large Toffoli gates from a quantum circuit while keeping the cumulative error low. We use the technique to give another proof that parity cannot be computed by constant depth quantum circuits without ancillæ. 1

Abstract. We define and construct efficient depth-universal and almostsize-universal quantum circuits. Such circuits can be viewed as generalpurpose simulators for central classes of quantum circuits and can be used to capture the computational power of the circuit class being simulated. For depth we construct universal circuits whose depth is the same order as the circuits being simulated. For size, there is a log factor blow-up in the universal circuits constructed here. We prove that this construction is nearly optimal. 1

We identify a sub-class of BQP that captures certain structural commonalities among many quantum algorithms including Shor’s algorithms. This class does not contain all of BQP (e.g. Grover’s algorithm does not fall into this class). Our main result is that any algorithm in this class that measures at most O(log n) qubits can be simulated by classical randomized polynomial time algorithms. This does not dequantize Shor’s algorithm (as the latter measures n qubits) but our work also highlights a new potentially hard function for cryptographic applications. Our main technical contribution is (to the best of our knowledge) a new exact characterization of certain sums of Fourier-type coefficients (with exponentially many summands). One of the key problems in complexity theory is to determine the power of the complexity class BQP. Recall that this is the set of languages accepted by uniform polynomial size quantum circuits with bounded two-sided error. It is essentially the quantum version of the complexity class BPP. Just as BPP corresponds to what is feasible on a classical computer with randomness, BQP corresponds

We show that, for any n> 0, the Heisenberg interaction among 2n qubits (as spin-1/2 particles) can be used to exactly implement an n-qubit parity gate, which is equivalent in constant depth to an n-qubit fanout gate. Either isotropic or nonisotropic versions of the interaction can be used. We generalize our basic results by showing that any Hamiltonian (acting on suitably encoded logical qubits), whose eigenvalues depend quadratically on the Hamming weight of the logical qubit values, can be used to implement generalized Modq gates for any q ≥ 2. This paper is a sequel to quant-ph/0309163, and resolves a question left open in that paper. 1