Stephen S. Bullock

Synthesis of quantum logic circuits

The pressure of fundamental limits on classical computation and the promise of exponential speedups from quantum effects have recently brought quantum circuits (Proc. R. Soc. Lond. A, Math. Phys. Sci., vol. 425, p. 73, 1989) to the attention of the electronic design automation community (Proc. 40th ACM/IEEE Design Automation Conf., 2003), (Phys. Rev. A, At. Mol. Opt. Phy., vol. 68, p. 012318, 2003), (Proc. 41st Design Automation Conf., 2004), (Proc. 39th Design Automation Conf., 2002), (Proc. Design, Automation, and Test Eur., 2004), (Phys. Rev. A, At. Mol. Opt. Phy., vol. 69, p. 062321, 2004), (IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., vol. 22, p. 710, 2003). Efficient quantum-logic circuits that perform two tasks are discussed: 1) implementing generic quantum computations, and 2) initializing quantum registers. In contrast to conventional computing, the latter task is nontrivial because the state space of an n-qubit register is not finite and contains exponential superpositions of classical bitstrings. The proposed circuits are asymptotically optimal for respective tasks and improve earlier published results by at least a factor of 2. The circuits for generic quantum computation constructed by the algorithms are the most efficient known today in terms of the number of most expensive gates [quantum controlled-NOTs (CNOTs)]. They are based on an analog of the Shannon decomposition of Boolean functions and a new circuit block, called quantum multiplexor (QMUX), which generalizes several known constructions. A theoretical lower bound implies that the circuits cannot be improved by more than a factor of 2. It is additionally shown how to accommodate the severe architectural limitation of using only nearest neighbor gates, which is representative of current implementation technologies. This increases the number of gates by almost an order of magnitude, but preserves the asymptotic optimality of gate counts

Minimal universal two-qubit controlled-NOT-based circuits

We give quantum circuits that simulate an arbitrary two-qubit unitary operator up to global phase. For several quantum gate libraries we prove that gate counts are optimal in worst and average cases. Our lower and upper bounds compare favorably to previously published results. Temporary storage is not used because it tends to be expensive in physical implementations. For each gate library, best gate counts can be achieved by a single universal circuit. To compute gate parameters in universal circuits, we only use closed-form algebraic expressions, and in particular do not rely on matrix exponentials. Our algorithm has been coded in C++.

Asymptotically Optimal Quantum Circuits for d-Level Systems

Scalability of a quantum computation requires that the information be processed on multiple subsystems. However, it is unclear how the complexity of a quantum algorithm, quantified by the number of entangling gates, depends on the subsystem size. We examine the quantum circuit complexity for exactly universal computation on many d-level systems (qudits). Both a lower bound and a constructive upper bound on the number of two-qudit gates result, proving a sharp asymptotic of theta(d(2n)) gates. This closes the complexity question for all d-level systems (d finite). The optimal asymptotic applies to systems with locality constraints, e.g., nearest neighbor interactions.

Recognizing small-circuit structure in two-qubit operators

This work proposes numerical tests which determine whether a two-qubit operator has an atypically simple quantum circuit. Specifically, we describe formulas, written in terms of matrix coefficients, characterizing operators implementable with exactly zero, one, or two controlled-NOT (CNOT) gates and all other gates being one-qubit gates. We give an algorithm for synthesizing two-qubit circuits with an optimal number of CNOT gates and illustrate it on operators appearing in quantum algorithms by Deutsch-Josza, Shor, and Grover. In another application, our explicit numerical tests allow timing a given Hamiltonian to compute a CNOT modulo one-qubit gate, when this is possible.

Canonical decompositions of n-qubit quantum computations and concurrence

The two-qubit canonical decomposition SU(4)=[SU(2)⊗SU(2)]Δ[SU(2)⊗SU(2)] writes any two-qubit unitary operator as a composition of a local unitary, a relative phasing of Bell states, and a second local unitary. Using Lie theory, we generalize this to an n-qubit decomposition, the concurrence canonical decomposition (CCD) SU(2n)=KAK. The group K fixes a bilinear form related to the concurrence, and in particular any unitary in K preserves the tangle |〈φ|¯(−iσ1y)⋯(−iσny)|φ〉|2 for n even. Thus, the CCD shows that any n-qubit unitary is a composition of a unitary operator preserving this n-tangle, a unitary operator in A which applies relative phases to a set of GHZ states, and a second unitary operator which preserves the tangle. As an application, we study the extent to which a large, random unitary may change concurrence. The result states that for a randomly chosen a∈A⊂SU(22p), the probability that a carries a state of tangle 0 to a state of maximum tangle approaches 1 as the even number of qubits approach...

Qudit surface codes and gauge theory with finite cyclic groups

Surface codes describe quantum memory stored as a global property of interacting spins on a surface. The state space is fixed by a complete set of quasi-local stabilizer operators and the code dimension depends on the first homology group of the surface complex. These code states can be actively stabilized by measurements or, alternatively, can be prepared by cooling to the ground subspace of a quasi-local spin Hamiltonian. In the case of spin-1/2 (qubit) lattices, such ground states have been proposed as topologically protected memory for qubits. We extend these constructions to lattices or more generally cell complexes with qudits, either of prime level or of level d? for d prime and ? ? 0, and therefore under tensor decomposition, to arbitrary finite levels. The Hamiltonian describes an exact gauge theory whose excitations correspond to Abelian anyons. We provide protocols for qudit storage and retrieval and propose an interferometric verification of topological order by measuring quasi-particle statistics.

Parallelism for quantum computation with qudits

Robust quantum computation with d-level quantum systems (qudits) poses two requirements: fast, parallel quantum gates and high-fidelity two-qudit gates. We first describe how to implement parallel single-qudit operations. It is by now well known that any single-qudit unitary can be decomposed into a sequence of Givens rotations on two-dimensional subspaces of the qudit state space. Using a coupling graph to represent physically allowed couplings between pairs of qudit states, we then show that the logical depth (time) of the parallel gate sequence is equal to the height of an associated tree. The implementation of a given unitary can then optimize the tradeoff between gate time and resources used. These ideas are illustrated for qudits encoded in the ground hyperfine states of the alkali-metal atoms {sup 87}Rb and {sup 133}Cs. Second, we provide a protocol for implementing parallelized nonlocal two-qudit gates using the assistance of entangled qubit pairs. Using known protocols for qubit entanglement purification, this offers the possibility of high-fidelity two-qudit gates.

Time reversal and n-qubit canonical decompositions

On pure states of n quantum bits, the concurrence entanglement monotone returns the norm of the inner product of a pure state with its spin-flip. The monotone vanishes for n odd, but for n even there is an explicit formula for its value on mixed states, i.e., a closed-form expression computes the minimum over all ensemble decompositions of a given density. For n even a matrix decomposition ν=k1ak2 of the unitary group is explicitly computable and allows for study of the monotone’s dynamics. The side factors k1 and k2 of this concurrence canonical decomposition (CCD) are concurrence symmetries, so the dynamics reduce to consideration of the a factor. This unitary a phases a basis of entangled states, and the concurrence dynamics of u are determined by these relative phases. In this work, we provide an explicit numerical algorithm computing ν=k1ak2 for n odd. Further, in the odd case we lift the monotone to a two-argument function. The concurrence capacity of ν according to the double argument lift may be n...

Smaller two-qubit circuits for quantum communication and computation

We show how to implement an arbitrary two-qubit unitary operation using any of several quantum gate libraries with small a priori upper bounds on gate counts. In analogy to library-less logic synthesis, we consider circuits and gates in terms of the underlying model of quantum computation, and do not assume any particular technology. As increasing the number of qubits can be prohibitively expensive, we assume throughout that no extra qubits are available for temporary storage. Using quantum circuit identities, we improve an earlier lower bound of 17 elementary gates by Bullock and Markov to 18, and their upper bound of 23 elementary gates to 18. We also improve upon the generic circuit with six CNOT gates by Zhang et al. (our circuit uses three), and that by Vidal and Dawson with 11 basic gates (we use 10). We study the performance of our synthesis procedures on two-qubit operators that are useful in quantum algorithms and communication protocols. With additional work, we find small circuits and improve upon previously known circuits in some cases.

Finding small two-qubit circuits

An important result from the mid nineties shows that any unitary evolution may be realized as a sequence of controlled-not and one-qubit gates. This work surveys especially efficient circuits in this library, in the special case of evolutions on two-quantum bits. In particular, we show that to construct an arbitrary two-qubit state from |00>, one CNOT gate suffices. To simulate an arbitrary two-qubit operator up to relative phases, two CNOTs suffice. To simulate an arbitrary two-qubit operator up to global phase, three CNOTs suffice. In each case, we construct an explicit circuit and prove optimality in the generic case. We also contribute a procedure to determine the minimal number of CNOT gates necessary to simulate a given two-qubit operator up to global phase. We use this procedure to discuss timing a given Hamiltonian to simulate the CNOT and to determine an optimal circuit for the two-qubit Quantum Fourier Transform. Our constructive proofs amount to circuit synthesis algorithms and have been coded in C++.