Michele Mosca

Quantum lower bounds by polynomials

We examine the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}/sup N/ in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions (i.e. problems involving a promise on the input) by Deutsch and Jozsa and by Simon cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with bounded-error using T black-box queries then there is a classical deterministic algorithm that computes f exactly with O(T/sup 6/) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.

Private quantum channels

We investigate how a classical private key can be used by two players, connected by an insecure one-way quantum channel, to perform private communication of quantum information. In particular, we show that in order to transmit n qubits privately, 2n bits of shared private key are necessary and sufficient. This result may be viewed as the quantum analogue of the classical one-time pad encryption scheme.

The Hidden Subgroup Problem and Eigenvalue Estimation on a Quantum Computer

A quantum computer can efficiently find the order of an element in a group, factors of composite integers, discrete logarithms, stabilisers in Abelian groups, and hidden or unknown subgroups of Abelian groups. It is already known how to phrase the first four problems as the estimation of eigenvalues of certain unitary operators. Here we show how the solution to the more general Abelian hidden subgroup problem can also be described and analysed as such. We then point out how certain instances of these problems can be solved with only one control qubit, or flying qubits, instead of entire registers of control qubits.

A Meet-in-the-Middle Algorithm for Fast Synthesis of Depth-Optimal Quantum Circuits

We present an algorithm for computing depth-optimal decompositions of logical operations, leveraging a meet-in-the-middle technique to provide a significant speedup over simple brute force algorithms. As an illustration of our method, we implemented this algorithm and found factorizations of commonly used quantum logical operations into elementary gates in the Clifford+T set. In particular, we report a decomposition of the Toffoli gate over the set of Clifford and T gates. Our decomposition achieves a total T-depth of 3, thereby providing a 40% reduction over the previously best known decomposition for the Toffoli gate. Due to the size of the search space, the algorithm is only practical for small parameters, such as the number of qubits, and the number of gates in an optimal implementation.

Quantum search on bounded-error inputs

Suppose we have n algorithms, quantum or classical, each computing some bit-value with bounded error probability. We describe a quantum algorithm that uses O(√n) repetitions of the base algorithms and with high probability finds the index of a 1-bit among these n bits (if there is such an index). This shows that it is not necessary to first significantly reduce the error probability in the base algorithms to O(1/poly(n)) (which would require O(√n log n) repetitions in total). Our technique is a recursive interleaving of amplitude amplification and error-reduction, and may be of more general interest. Essentially, it shows that quantum amplitude amplification can be made to work also with a bounded-error verifier. As a corollary we obtain optimal quantum upper bounds of O(√N) queries for all constant-depth AND-OR trees on N variables, improving upon earlier upper bounds of O(√Npolylog(N)).

Approximate quantum cloning with nuclear magnetic resonance.

Here we describe a nuclear magnetic resonance (NMR) experiment that uses a three qubit NMR device to implement the one-to-two approximate quantum cloning network of Buzek et al. [Phys. Rev. A 56, 3446 (1997)]. As expected the experimental results indicate that the network clones all input states with similar fidelities, but as a result of decoherence and incoherent evolution arising from B(1) inhomogeneity the total fidelity achieved does not exceed the measurement bound.

Asymptotically optimal approximation of single qubit unitaries by Clifford and T circuits using a constant number of ancillary qubits.

Decomposing unitaries into a sequence of elementary operations is at the core of quantum computing. Information theoretic arguments show that approximating a random unitary with precision ε requires Ω(log(1/ε)) gates. Prior to our work, the state of the art in approximating a single qubit unitary included the Solovay-Kitaev algorithm that requires O(log(3+δ)(1/ε)) gates and does not use ancillae and the phase kickback approach that requires O(log(2)(1/ε)loglog(1/ε)) gates but uses O(log(2)(1/ε)) ancillae. Both algorithms feature upper bounds that are far from the information theoretic lower bound. In this Letter, we report an algorithm that saturates the lower bound, and as such it guarantees asymptotic optimality. In particular, we present an algorithm for building a circuit that approximates single qubit unitaries with precision ε using O(log(1/ε)) Clifford and T gates and employing up to two ancillary qubits. We connect the unitary approximation problem to the problem of constructing solutions corresponding to Lagranges four-square theorem, and thereby develop an algorithm for computing an approximating circuit using an average of O(log(2)(1/ε)loglog(1/ε)) operations with integers.

Quantum Logic Gates and Nuclear Magnetic Resonance Pulse Sequences

There has recently been considerable interest in the use of nuclear magnetic resonance (NMR) as a technology for the implementation of small quantum computers. These computers operate by the laws of quantum mechanics, rather than classical mechanics and can be used to implement new quantum algorithms. Here we describe how NMR in principle can be used to implement all the elements required to build quantum computers, and draw comparisons between the pulse sequences involved and those of more conventional NMR experiments.

Polynomial-Time T-Depth Optimization of Clifford+T Circuits Via Matroid Partitioning

Most work in quantum circuit optimization has been performed in isolation from the results of quantum fault-tolerance. Here we present a polynomial-time algorithm for optimizing quantum circuits that takes the actual implementation of fault-tolerant logical gates into consideration. Our algorithm resynthesizes quantum circuits composed of Clifford group and T gates, the latter being typically the most costly gate in fault-tolerant models, e.g., those based on the Steane or surface codes, with the purpose of minimizing both T-count and T-depth. A major feature of the algorithm is the ability to resynthesize circuits with ancillae at effectively no additional cost, allowing space-time trade-offs to be easily explored. The tested benchmarks show up to 65.7% reduction in T-count and up to 87.6% reduction in T-depth without ancillae, or 99.7% reduction in T-depth using ancillae.

APPROXIMATE QUANTUM COUNTING ON AN NMR ENSEMBLE QUANTUM COMPUTER

We demonstrate the implementation of a quantum algorithm for estimating the number of matching items in a search operation using a two qubit nuclear magnetic resonance (NMR) quantum computer.