Classical Rules in Quantum Games

We consider two aspects of quantum game theory: the extent to which the quantum solution solves the original classical game, and to what extent the new solution can be obtained in a classical model.Comment: The previous title, "Quantum games are no fun (yet)", was too whimsical for Physical Review. This is a comment on most, but not all, papers on quantum game theor

Restrictions on Transversal Encoded Quantum Gate Sets

Transversal gates play an important role in the theory of fault-tolerant quantum computation due to their simplicity and robustness to noise. By definition, transversal operators do not couple physical subsystems within the same code block. Consequently, such operators do not spread errors within code blocks and are, therefore, fault tolerant. Nonetheless, other methods of ensuring fault tolerance are required, as it is invariably the case that some encoded gates cannot be implemented transversally. This observation has led to a long-standing conjecture that transversal encoded gate sets cannot be universal. Here we show that the ability of a quantum code to detect an arbitrary error on any single physical subsystem is incompatible with the existence of a universal, transversal encoded gate set for the code.Comment: 4 pages, v2: minor change

Quantum Error Correction and Orthogonal Geometry

A group theoretic framework is introduced that simplifies the description of known quantum error-correcting codes and greatly facilitates the construction of new examples. Codes are given which map 3 qubits to 8 qubits correcting 1 error, 4 to 10 qubits correcting 1 error, 1 to 13 qubits correcting 2 errors, and 1 to 29 qubits correcting 5 errors.Comment: RevTex, 4 pages, no figures, submitted to Phys. Rev. Letters. We have changed the statement of Theorem 2 to correct it -- we now get worse rates than we previously claimed for our quantum codes. Minor changes have been made to the rest of the pape

Quantum Error Correction via Codes over GF(4)

The problem of finding quantum error-correcting codes is transformed into the problem of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.Comment: Latex, 46 pages. To appear in IEEE Transactions on Information Theory. Replaced Sept. 24, 1996, to correct a number of minor errors. Replaced Sept. 10, 1997. The second section has been completely rewritten, and should hopefully be much clearer. We have also added a new section discussing the developments of the past year. Finally, we again corrected a number of minor error

Magnetic qubits as hardware for quantum computers

We propose two potential realisations for quantum bits based on nanometre scale magnetic particles of large spin S and high anisotropy molecular clusters. In case (1) the bit-value basis states |0> and |1> are the ground and first excited spin states Sz = S and S-1, separated by an energy gap given by the ferromagnetic resonance (FMR) frequency. In case (2), when there is significant tunnelling through the anisotropy barrier, the qubit states correspond to the symmetric, |0>, and antisymmetric, |1>, combinations of the two-fold degenerate ground state Sz = +- S. In each case the temperature of operation must be low compared to the energy gap, \Delta, between the states |0> and |1>. The gap \Delta in case (2) can be controlled with an external magnetic field perpendicular to the easy axis of the molecular cluster. The states of different molecular clusters and magnetic particles may be entangled by connecting them by superconducting lines with Josephson switches, leading to the potential for quantum computing hardware.Comment: 17 pages, 3 figure

Unified derivations of measurement-based schemes for quantum computation

We present unified, systematic derivations of schemes in the two known measurement-based models of quantum computation. The first model (introduced by Raussendorf and Briegel [Phys. Rev. Lett., 86, 5188 (2001)]) uses a fixed entangled state, adaptive measurements on single qubits, and feedforward of the measurement results. The second model (proposed by Nielsen [Phys. Lett. A, 308, 96 (2003)] and further simplified by Leung [Int. J. Quant. Inf., 2, 33 (2004)]) uses adaptive two-qubit measurements that can be applied to arbitrary pairs of qubits, and feedforward of the measurement results. The underlying principle of our derivations is a variant of teleportation introduced by Zhou, Leung, and Chuang [Phys. Rev. A, 62, 052316 (2000)]. Our derivations unify these two measurement-based models of quantum computation and provide significantly simpler schemes.Comment: 14 page

Quantum Communication With Zero-Capacity Channels

Communication over a noisy quantum channel introduces errors in the transmission that must be corrected. A fundamental bound on quantum error correction is the quantum capacity, which quantifies the amount of quantum data that can be protected. We show theoretically that two quantum channels, each with a transmission capacity of zero, can have a nonzero capacity when used together. This unveils a rich structure in the theory of quantum communications, implying that the quantum capacity does not uniquely specify a channel's ability for transmitting quantum information.Comment: Final submitted versio

A Study on the Noise Threshold of Fault-tolerant Quantum Error Correction

Quantum circuits implementing fault-tolerant quantum error correction (QEC) for the three qubit bit-flip code and five-qubit code are studied. To describe the effect of noise, we apply a model based on a generalized effective Hamiltonian where the system-environment interactions are taken into account by including stochastic fluctuating terms in the system Hamiltonian. This noise model enables us to investigate the effect of noise in quantum circuits under realistic device conditions and avoid strong assumptions such as maximal parallelism and weak storage errors. Noise thresholds of the QEC codes are calculated. In addition, the effects of imprecision in projective measurements, collective bath, fault-tolerant repetition protocols, and level of parallelism in circuit constructions on the threshold values are also studied with emphasis on determining the optimal design for the fault-tolerant QEC circuit. These results provide insights into the fault-tolerant QEC process as well as useful information for designing the optimal fault-tolerant QEC circuit for particular physical implementation of quantum computer.Comment: 9 pages, 9 figures; to be submitted to Phys. Rev.

Time complexity and gate complexity

We formulate and investigate the simplest version of time-optimal quantum computation theory (t-QCT), where the computation time is defined by the physical one and the Hamiltonian contains only one- and two-qubit interactions. This version of t-QCT is also considered as optimality by sub-Riemannian geodesic length. The work has two aims: one is to develop a t-QCT itself based on physically natural concept of time, and the other is to pursue the possibility of using t-QCT as a tool to estimate the complexity in conventional gate-optimal quantum computation theory (g-QCT). In particular, we investigate to what extent is true the statement: time complexity is polynomial in the number of qubits if and only if so is gate complexity. In the analysis, we relate t-QCT and optimal control theory (OCT) through fidelity-optimal computation theory (f-QCT); f-QCT is equivalent to t-QCT in the limit of unit optimal fidelity, while it is formally similar to OCT. We then develop an efficient numerical scheme for f-QCT by modifying Krotov's method in OCT, which has monotonic convergence property. We implemented the scheme and obtained solutions of f-QCT and of t-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 exponential one because a series of generic unitary operators has a 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. Thus the both targets are examples satisfyng the statement above. 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.Comment: 11 pages, 6 figure