Quantum Computing

Quantum computers may offer a way to solve certain problems that are larger and more complex than any that can be solved on conventional computers. One way to view a (gated) quantum computer is as a machine that multiplies a vector by a unitary matrix. The number of possible data values equals the dimension of the vector, and the absolute value of the $i$th component of the vector represents the probability that the data or the answer is equal to the $i$th value. The unitary matrix is designed to make the absolute value of the entry corresponding to the correct answer quite close to one.

Stephen Bullock, Gavin Brennen, and I investigated several questions relevant to the decomposition of the unitary matrix into quantum ``gates" that can actually be implemented in hardware. In [J69], we used a matrix decomposition to construct these gates. In [J70], [J71] and [J75], we used controlled Householder gates to implement circuits for qudits, quantum variables that can take on $d>2$ values rather than the traditional 2 values used for qubits. Some Givens and Householder gates are cheaper than others, depending on the components they access, and we determined a systematic way to determine whether a set of rotation planes is sufficient in [J72]. Then we considered how much such quantum operations might be sped up [J78], for example by using more than one pair of lasers to excite multiple transitions among the hyperfine states of the atomic alkalies.

Another possible mechanism for quantum computing is the use of adiabatic systems. In [J86], Michael O'Hara and I investigated the effects of perturbations on such systems.

Finally, O'Hara and I studied the unexpectedly large ground-state energy gaps in a certain class of Hamiltonians [J92].

[J69]

Stephen S. Bullock, Gavin K. Brennen, and Dianne P. O'Leary, ``Time reversal and $n$-qubit Canonical Decompositions," Journal of Mathematical Physics 46, 062104 (2005) 18 pages. http://xxx.lanl.gov/abs/quant-ph/0402051

[J70]

Gavin K. Brennen, Dianne P. O'Leary, and Stephen S. Bullock, ``Criteria for Exact Qudit Universality," Physical Review A 71, 052318 (2005) 7 pages. http://xxx.lanl.gov/abs/quant-ph/0407223

[J71]

Stephen S. Bullock, Dianne P. O'Leary, and Gavin K. Brennen, ``Asymptotically optimal quantum circuits for $d$-level systems," Physical Review Letters 94, no 23 (2005) 230502, 4 pages. http://xxx.lanl.gov/abs/quant-ph/0410116

[J72]

Dianne P. O'Leary and Stephen S. Bullock, ``QR Factorizations Using a Restricted Set of Rotations," Electronic Transactions on Numerical Analysis, 21 (2005) pp. 20-27. http://etna.mcs.kent.edu/

[J75]

Gavin K. Brennen, Stephen S. Bullock, and Dianne P. O'Leary, ``Efficient Circuits for Exact-Universal Computation with Qudits," Quantum Information and Computation, 6 (2006), 436-454.

[J78]

Dianne P. O'Leary, Gavin K. Brennen, and Stephen S. Bullock, ``Parallelism for Quantum Computation with Qudits," Physical Review A, (2006) 74:3, doi:10.1103/PhysRevA.74.032334

[J86]

Michael J. O'Hara and Dianne P. O'Leary, ``The Adiabatic Theorem in the Presence of Noise," Physical Review A, 77 (2008) 042319, 20 pages. http://link.aps.org/abstract/PRA/v77/e042319DOI: 10.1103/PhysRevA.77.042319. Chosen for inclusion in Virtual Journal of Applications of Superconductivity 14:9 (2008) and Virtual Journal of Quantum Information (2008).

[J92]

Michael J. O'Hara and Dianne P. O'Leary, ``Quadratic Fermionic Interactions Yield Hamiltonians with Large Ground-State Energy Gaps," Physical Review A, 79:3 (2009) 032331 (10 pages). DOI: 10.1103/PhysRevA.79.032331 http://link.aps.org/abstract/PRA/v79/e032331Chosen for inclusion in Virtual Journal of Quantum Information 9:4 (2009).