Numerical simulation of information recovery in quantum computers

Decoherence is the main problem to be solved before quantum computers can be built. To control decoherence, it is possible to use error correction methods, but these methods are themselves noisy quantum computation processes. In this work we study the ability of Steane's and Shor's fault-tolerant recovering methods, as well a modification of Steane's ancilla network, to correct errors in qubits. We test a way to measure correctly ancilla's fidelity for these methods, and state the possibility of carrying out an effective error correction through a noisy quantum channel, even using noisy error correction methods.Comment: 38 pages, Figures included. Accepted in Phys. Rev. A, 200

Scalability of Shor's algorithm with a limited set of rotation gates

Typical circuit implementations of Shor's algorithm involve controlled rotation gates of magnitude $\pi/2^{2L}$ where $L$ is the binary length of the integer N to be factored. Such gates cannot be implemented exactly using existing fault-tolerant techniques. Approximating a given controlled $\pi/2^{d}$ rotation gate to within $\delta=O(1/2^{d})$ currently requires both a number of qubits and number of fault-tolerant gates that grows polynomially with $d$. In this paper we show that this additional growth in space and time complexity would severely limit the applicability of Shor's algorithm to large integers. Consequently, we study in detail the effect of using only controlled rotation gates with $d$ less than or equal to some $d_{\rm max}$. It is found that integers up to length $L_{\rm max} = O(4^{d_{\rm max}})$ can be factored without significant performance penalty implying that the cumbersome techniques of fault-tolerant computation only need to be used to create controlled rotation gates of magnitude $\pi/64$ if integers thousands of bits long are desired factored. Explicit fault-tolerant constructions of such gates are also discussed.Comment: Substantially revised version, twice as long as original. Two tables converted into one 8-part figure, new section added on the construction of arbitrary single-qubit rotations using only the fault-tolerant gate set. Substantial additional discussion and explanatory figures added throughout. (8 pages, 6 figures

Simple Quantum Error Correcting Codes

Methods of finding good quantum error correcting codes are discussed, and many example codes are presented. The recipe C_2^{\perp} \subseteq C_1, where C_1 and C_2 are classical codes, is used to obtain codes for up to 16 information qubits with correction of small numbers of errors. The results are tabulated. More efficient codes are obtained by allowing C_1 to have reduced distance, and introducing sign changes among the code words in a systematic manner. This systematic approach leads to single-error correcting codes for 3, 4 and 5 information qubits with block lengths of 8, 10 and 11 qubits respectively.Comment: Submitted to Phys. Rev. A. in May 1996. 21 pages, no figures. Further information at http://eve.physics.ox.ac.uk/ASGhome.htm

Effects of noise on quantum error correction algorithms

It has recently been shown that there are efficient algorithms for quantum computers to solve certain problems, such as prime factorization, which are intractable to date on classical computers. The chances for practical implementation, however, are limited by decoherence, in which the effect of an external environment causes random errors in the quantum calculation. To combat this problem, quantum error correction schemes have been proposed, in which a single quantum bit (qubit) is ``encoded'' as a state of some larger number of qubits, chosen to resist particular types of errors. Most such schemes are vulnerable, however, to errors in the encoding and decoding itself. We examine two such schemes, in which a single qubit is encoded in a state of $n$ qubits while subject to dephasing or to arbitrary isotropic noise. Using both analytical and numerical calculations, we argue that error correction remains beneficial in the presence of weak noise, and that there is an optimal time between error correction steps, determined by the strength of the interaction with the environment and the parameters set by the encoding.Comment: 26 pages, LaTeX, 4 PS figures embedded. Reprints available from the authors or http://eve.physics.ox.ac.uk/QChome.htm

Search for correlation effects in linear chains of trapped ions

We report a precise search for correlation effects in linear chains of 2 and 3 trapped Ca+ ions. Unexplained correlations in photon emission times within a linear chain of trapped ions have been reported, which, if genuine, cast doubt on the potential of an ion trap to realize quantum information processing. We observe quantum jumps from the metastable 3d 2D_{5/2} level for several hours, searching for correlations between the decay times of the different ions. We find no evidence for correlations: the number of quantum jumps with separations of less than 10 ms is consistent with statistics to within errors of 0.05%; the lifetime of the metastable level derived from the data is consistent with that derived from independent single-ion data at the level of the experimental errors 1%; and no rank correlations between the decay times were found with sensitivity to rank correlation coefficients at the level of |R| = 0.024.Comment: With changes to introduction. 5 pages, including 4 figures. Submitted to Europhys. Let

Quantum Analogue Computing

We briefly review what a quantum computer is, what it promises to do for us, and why it is so hard to build one. Among the first applications anticipated to bear fruit is quantum simulation of quantum systems. While most quantum computation is an extension of classical digital computation, quantum simulation differs fundamentally in how the data is encoded in the quantum computer. To perform a quantum simulation, the Hilbert space of the system to be simulated is mapped directly onto the Hilbert space of the (logical) qubits in the quantum computer. This type of direct correspondence is how data is encoded in a classical analogue computer. There is no binary encoding, and increasing precision becomes exponentially costly: an extra bit of precision doubles the size of the computer. This has important consequences for both the precision and error correction requirements of quantum simulation, and significant open questions remain about its practicality. It also means that the quantum version of analogue computers, continuous variable quantum computers (CVQC) becomes an equally efficient architecture for quantum simulation. Lessons from past use of classical analogue computers can help us to build better quantum simulators in future.Comment: 10 pages, to appear in the Visions 2010 issue of Phil. Trans. Roy. Soc.

Decoherence of geometric phase gates

We consider the effects of certain forms of decoherence applied to both adiabatic and non-adiabatic geometric phase quantum gates. For a single qubit we illustrate path-dependent sensitivity to anisotropic noise and for two qubits we quantify the loss of entanglement as a function of decoherence.Comment: 4 pages, 3 figure

Speed of ion trap quantum information processors

We investigate theoretically the speed limit of quantum gate operations for ion trap quantum information processors. The proposed methods use laser pulses for quantum gates which entangle the electronic and vibrational degrees of freedom of the trapped ions. Two of these methods are studied in detail and for both of them the speed is limited by a combination of the recoil frequency of the relevant electronic transition, and the vibrational frequency in the trap. We have experimentally studied the gate operations below and above this speed limit. In the latter case, the fidelity is reduced, in agreement with our theoretical findings. // Changes: a) error in equ. 24 and table III repaired b) reference Jonathan et al, quant-ph/ 0002092, added (proposes fast quantum gates using the AC-Stark effect)Comment: 10 pages, 4 figure

Prevention of dissipation with two particles

An error prevention procedure based on two-particle encoding is proposed for protecting an arbitrary unknown quantum state from dissipation, such as phase damping and amplitude damping. The schemes, which exhibits manifestation of the quantum Zeno effect, is effective whether quantum bits are decohered independently or cooperatively. We derive the working condition of the scheme and argue that this procedure has feasible practical implementation.Comment: 12 pages, Late