Abstract
We formulate and investigate the simplest version of time-optimal quantum computation theory (TO-QCT), where the computation time is defined by the physical one and the Hamiltonian contains only one- and two-qubit interactions. This version of TO-QCT is also considered as optimality by sub-Riemannian geodesic length. The work has two aims: One is to develop a TO-QCT itself based on a physically natural concept of time, and the other is to pursue the possibility of using TO-QCT as a tool to estimate the complexity in conventional gate-optimal quantum computation theory (GO-QCT). In particular, we investigate to what extent is true the following statement: Time complexity is polynomial in the number of qubits if and only if gate complexity is also. In the analysis, we relate TO-QCT and optimal control theory (OCT) through fidelity-optimal computation theory (FO-QCT); FO-QCT is equivalent to TO-QCT in the limit of unit optimal fidelity, while it is formally similar to OCT. We then develop an efficient numerical scheme for FO-QCT by modifying Krotov's method in OCT, which has a monotonic convergence property. We implemented the scheme and obtained solutions of FO-QCT and of TO-QCT for the quantum Fourier transform and a unitary operator that does not have an apparent symmetry. The former has a polynomial gate complexity and the latter is expected to have an exponential one which is based on the fact that a series of generic unitary operators has an exponential gate complexity. The time complexity for the former is found to be linear in the number of qubits, which is understood naturally by the existence of an upper bound. The time complexity for the latter is exponential in the number of qubits. Thus, both the targets seem to be examples satisfyng the preceding statement. The typical characteristics of the optimal Hamiltonians are symmetry under time reversal and constancy of one-qubit operation, which are mathematically shown to hold in fairly general situations.
| Original language | English |
|---|---|
| Article number | 042305 |
| Journal | Physical Review A - Atomic, Molecular, and Optical Physics |
| Volume | 82 |
| Issue number | 4 |
| DOIs | |
| Publication status | Published - 2010 Oct 7 |
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ASJC Scopus subject areas
- Atomic and Molecular Physics, and Optics
Cite this
Time complexity and gate complexity. / Koike, Tatsuhiko; Okudaira, Yosuke.
In: Physical Review A - Atomic, Molecular, and Optical Physics, Vol. 82, No. 4, 042305, 07.10.2010.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Time complexity and gate complexity
AU - Koike, Tatsuhiko
AU - Okudaira, Yosuke
PY - 2010/10/7
Y1 - 2010/10/7
N2 - We formulate and investigate the simplest version of time-optimal quantum computation theory (TO-QCT), where the computation time is defined by the physical one and the Hamiltonian contains only one- and two-qubit interactions. This version of TO-QCT is also considered as optimality by sub-Riemannian geodesic length. The work has two aims: One is to develop a TO-QCT itself based on a physically natural concept of time, and the other is to pursue the possibility of using TO-QCT as a tool to estimate the complexity in conventional gate-optimal quantum computation theory (GO-QCT). In particular, we investigate to what extent is true the following statement: Time complexity is polynomial in the number of qubits if and only if gate complexity is also. In the analysis, we relate TO-QCT and optimal control theory (OCT) through fidelity-optimal computation theory (FO-QCT); FO-QCT is equivalent to TO-QCT in the limit of unit optimal fidelity, while it is formally similar to OCT. We then develop an efficient numerical scheme for FO-QCT by modifying Krotov's method in OCT, which has a monotonic convergence property. We implemented the scheme and obtained solutions of FO-QCT and of TO-QCT for the quantum Fourier transform and a unitary operator that does not have an apparent symmetry. The former has a polynomial gate complexity and the latter is expected to have an exponential one which is based on the fact that a series of generic unitary operators has an exponential gate complexity. The time complexity for the former is found to be linear in the number of qubits, which is understood naturally by the existence of an upper bound. The time complexity for the latter is exponential in the number of qubits. Thus, both the targets seem to be examples satisfyng the preceding statement. The typical characteristics of the optimal Hamiltonians are symmetry under time reversal and constancy of one-qubit operation, which are mathematically shown to hold in fairly general situations.
AB - We formulate and investigate the simplest version of time-optimal quantum computation theory (TO-QCT), where the computation time is defined by the physical one and the Hamiltonian contains only one- and two-qubit interactions. This version of TO-QCT is also considered as optimality by sub-Riemannian geodesic length. The work has two aims: One is to develop a TO-QCT itself based on a physically natural concept of time, and the other is to pursue the possibility of using TO-QCT as a tool to estimate the complexity in conventional gate-optimal quantum computation theory (GO-QCT). In particular, we investigate to what extent is true the following statement: Time complexity is polynomial in the number of qubits if and only if gate complexity is also. In the analysis, we relate TO-QCT and optimal control theory (OCT) through fidelity-optimal computation theory (FO-QCT); FO-QCT is equivalent to TO-QCT in the limit of unit optimal fidelity, while it is formally similar to OCT. We then develop an efficient numerical scheme for FO-QCT by modifying Krotov's method in OCT, which has a monotonic convergence property. We implemented the scheme and obtained solutions of FO-QCT and of TO-QCT for the quantum Fourier transform and a unitary operator that does not have an apparent symmetry. The former has a polynomial gate complexity and the latter is expected to have an exponential one which is based on the fact that a series of generic unitary operators has an exponential gate complexity. The time complexity for the former is found to be linear in the number of qubits, which is understood naturally by the existence of an upper bound. The time complexity for the latter is exponential in the number of qubits. Thus, both the targets seem to be examples satisfyng the preceding statement. The typical characteristics of the optimal Hamiltonians are symmetry under time reversal and constancy of one-qubit operation, which are mathematically shown to hold in fairly general situations.
UR - http://www.scopus.com/inward/record.url?scp=77957778837&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=77957778837&partnerID=8YFLogxK
U2 - 10.1103/PhysRevA.82.042305
DO - 10.1103/PhysRevA.82.042305
M3 - Article
AN - SCOPUS:77957778837
VL - 82
JO - Physical Review A
JF - Physical Review A
SN - 2469-9926
IS - 4
M1 - 042305
ER -