Isaac L. Chuang

Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance

The number of steps any classical computer requires in order to find the prime factors of an l-digit integer N increases exponentially with l, at least using algorithms known at present. Factoring large integers is therefore conjectured to be intractable classically, an observation underlying the security of widely used cryptographic codes. Quantum computers, however, could factor integers in only polynomial time, using Shors quantum factoring algorithm. Although important for the study of quantum computers, experimental demonstration of this algorithm has proved elusive. Here we report an implementation of the simplest instance of Shors algorithm: factorization of N = 15 (whose prime factors are 3 and 5). We use seven spin-1/2 nuclei in a molecule as quantum bits, which can be manipulated with room temperature liquid-state nuclear magnetic resonance techniques. This method of using nuclei to store quantum information is in principle scalable to systems containing many quantum bits, but such scalability is not implied by the present work. The significance of our work lies in the demonstration of experimental and theoretical techniques for precise control and modelling of complex quantum computers. In particular, we present a simple, parameter-free but predictive model of decoherence effects in our system.

Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations

We present a method to create a variety of interesting gates by teleporting quantum bits through special entangled states. This allows, for instance, the construction of a quantum computer based on just single qubit operations, Bell measurements, and GHZ states. We also present straightforward constructions of a wide variety of fault-tolerant quantum gates.Algorithms such as quantum factoring and quantum search illustrate the great theoretical promise of quantum computers; but the practical implementation of such devices will require careful consideration of the minimum resource requirements, together with the development of procedures to overcome inevitable residual imperfections in physical systems. Many designs have been proposed, but none allow a large quantum computer to be built in the near future. Moreover, the known protocols for constructing reliable quantum computers from unreliable components can be complicated, often requiring many operations to produce a desired transformation. Here we show how a single technique—a generalization of quantum teleportation—reduces resource requirements for quantum computers and unifies known protocols for fault-tolerant quantum computation. We show that single quantum bit (qubit) operations, Bell-basis measurements and certain entangled quantum states such as Greenberger–Horne–Zeilinger (GHZ) states—all of which are within the reach of current technology—are sufficient to construct a universal quantum computer. We also present systematic constructions for an infinite class of reliable quantum gates that make the design of fault-tolerant quantum computers much more straightforward and methodical.

Decoherence free subspaces for quantum computation

Decoherence in quantum computers is formulated within the semigroup approach. The error generators are identified with the generators of a Lie algebra. This allows for a comprehensive description which includes as a special case the frequently assumed spin-boson model. A generic condition is presented for errorless quantum computation: decoherence-free subspaces are spanned by those states which are annihilated by all the generators. It is shown that these subspaces are stable to perturbations and, moreover, that universal quantum computation is possible within them.

NMR techniques for quantum control and computation

Fifty years of developments in nuclear magnetic resonance (NMR) have resulted in an unrivaled degree of control of the dynamics of coupled two-level quantum systems. This coherent control of nuclear spin dynamics has recently been taken to a new level, motivated by the interest in quantum information processing. NMR has been the workhorse for the experimental implementation of quantum protocols, allowing exquisite control of systems up to seven qubits in size. This article surveys and summarizes a broad variety of pulse control and tomographic techniques which have been developed for, and used in, NMR quantum computation. Many of these will be useful in other quantum systems now being considered for the implementation of quantum information processing tasks.

Experimental realization of a quantum algorithm

Quantum computers can in principle exploit quantum-mechanical effects to perform computations (such as factoring large numbers or searching an unsorted database) more rapidly than classical computers,,. But noise, loss of coherence, and manufacturing problems make constructing large-scale quantum computers difficult. Although ion traps and optical cavities offer promising experimental approaches,, no quantum algorithm has yet been implemented with these systems. Here we report the experimental realization of a quantum algorithm using a bulk nuclear magnetic resonance technique, in which the nuclear spins act as ‘quantum bits’. The nuclear spins are particularly suited to this role because of their natural isolation from the environment. Our simple quantum computer solves a purely mathematical problem in fewer steps than is possible classically, requiring fewer ‘function calls’ than a classical computer to determine the global properties of an unknown function.

Prescription for experimental determination of the dynamics of a quantum black box

Abstract We give an explicit way to experimentally determine the evolution operators which completely describe the dynamics of a quantum-mechanical black box: an arbitrary open quantum system. We show necessary and sufficient conditions for this to be possible and illustrate the general theory by considering specifically one-and two-quantum-bit systems. These procedures may be useful in the comparative evaluation of experimental quantum measurement, communication and computation systems.

Implementation of the Deutsch–Jozsa algorithm on an ion-trap quantum computer

Determining classically whether a coin is fair (head on one side, tail on the other) or fake (heads or tails on both sides) requires an examination of each side. However, the analogous quantum procedure (the Deutsch–Jozsa algorithm) requires just one examination step. The Deutsch–Jozsa algorithm has been realized experimentally using bulk nuclear magnetic resonance techniques, employing nuclear spins as quantum bits (qubits). In contrast, the ion trap processor utilises motional and electronic quantum states of individual atoms as qubits, and in principle is easier to scale to many qubits. Experimental advances in the latter area include the realization of a two-qubit quantum gate, the entanglement of four ions, quantum state engineering and entanglement-enhanced phase estimation. Here we exploit techniques developed for nuclear magnetic resonance to implement the Deutsch–Jozsa algorithm on an ion-trap quantum processor, using as qubits the electronic and motional states of a single calcium ion. Our ion-based implementation of a full quantum algorithm serves to demonstrate experimental procedures with the quality and precision required for complex computations, confirming the potential of trapped ions for quantum computation.

SIMPLE QUANTUM COMPUTER

We propose an implementation of a quantum computer to solve Deutschs problem, which requires exponential time on a classical computer but only linear time with quantum parallelism. By using a dual-rail quantum-bit representation as a simple form of error correction, our machine can tolerate some amount of decoherence and still give the correct result with high probability. The design that we employ also demonstrates a signature for quantum parallelism which unambiguously distinguishes the desired quantum behavior from the merely classical. The experimental demonstration of our proposal using quantum optical components calls for the development of several key technologies common to single photonics.

Programmable quantum gate arrays

We show how to construct quantum gate arrays that can be programmed to perform different unitary operations on a data register, depending on the input to some program register. It is shown that a universal quantum gate array\char22{}a gate array which can be programmed to perform any unitary operation\char22{}exists only if one allows the gate array to operate in a probabilistic fashion. Thus it is not possible to build a fixed, general purpose quantum computer which can be programmed to perform an arbitrary quantum computation.

Quantum Computers, Factoring, and Decoherence

It is known that quantum computers can dramatically speed up the task of finding factors of large numbers, a problem of practical significance for cryptographic applications. Factors of an L-digit number can be found in ∼L2 time [compared to ∼exp(L1/3) time] by a quantum computer, which simultaneously follows all paths corresponding to distinct classical inputs, obtaining the solution from the coherent quantum interference of the alternatives. Here it is shown how the decoherence process degrades the interference pattern that emerges from the quantum factoring algorithm. For a quantum computer performing logical operations, an exponential decay of quantum coherence is inevitable. However, even in the presence of exponential decoherence, quantum computation can be useful as long as a sufficiently low decoherence rate can be achieved to allow meaningful results to be extracted from the calculation.