Farrokh Vatan

Algorithmic cooling and scalable NMR quantum computers

We present here algorithmic cooling (via polarization heat bath)—a powerful method for obtaining a large number of highly polarized spins in liquid nuclear-spin systems at finite temperature. Given that spin-half states represent (quantum) bits, algorithmic cooling cleans dirty bits beyond the Shannons bound on data compression, by using a set of rapidly thermal-relaxing bits. Such auxiliary bits could be implemented by using spins that rapidly get into thermal equilibrium with the environment, e.g., electron spins. Interestingly, the interaction with the environment, usually a most undesired interaction, is used here to our benefit, allowing a cooling mechanism. Cooling spins to a very low temperature without cooling the environment could lead to a breakthrough in NMR experiments, and our “spin-refrigerating” method suggests that this is possible. The scaling of NMR ensemble computers is currently one of the main obstacles to building larger-scale quantum computing devices, and our spin-refrigerating method suggests that this problem can be resolved.

A new universal and fault-tolerant quantum basis

A novel universal and fault-tolerant basis (set of gates) for quantum computation is described. Such a set is necessary to perform quantum computation in a realistic noisy environment. The new basis consists only of two single-qubit gates (Hadamard and 1=4 z ), and one two-qubit gate (Controlled-NOT). Moreover, a new general method for fault-tolerant implementation of quantum gates like Toffoli is introduced. This method is a generalization of the methods suggested by Shor (Proc. FOCS’96, 1996, p. 56) and later by Knill et al. (Proc. Roy. Soc. London Ser. A 454 (1998) 365).

On universal and fault-tolerant quantum computing: a novel basis and a new constructive proof of universality for Shor's basis

A novel universal and fault-tolerant basis (set of gates) for quantum computation is described. Such a set is necessary to perform quantum computation in a realistic noisy environment. The new basis consists of two single-qubit gates (Hadamard and /spl sigma//sub z//sup 1/4 /) and one double-qubit gate (Controlled-NOT). Since the set consisting of Controlled-NOT and Hadamard gates is not universal, the new basis achieves universality by including only one additional elementary (in the sense that it does not include angles that are irrational multiples of /spl pi/) single-qubit gate, and hence, is potentially the simplest universal basis that one can construct. We also provide an alternative proof of universality for the only other known class of universal and fault-tolerant basis proposed by P.W. Shor (1996) and A.Y. Kitaev (1997).

Spatially correlated qubit errors and burst-correcting quantum codes

We explore the design of quantum error-correcting codes for cases where the decoherence events of qubits are correlated. In particular, we consider the case where only spatially contiguous qubits decohere, which is analogous to the case of burst errors in classical coding theory. We present several different efficient schemes for constructing families of such codes. For example, one can find one-dimensional quantum codes of length n=13 and 15 that correct burst errors of width b<3; as a comparison, a random-error correcting quantum code that corrects t=3 errors must have length n/spl ges/17. In general, we show that it is possible to build quantum burst-correcting codes that have near optimal dimension. For example, we show that for any constant b, there exist b-burst-correcting quantum codes with length n, and dimension k=n-log n-O(1); as a comparison, the Hamming bound for the case with t (constant) random errors yields k/spl les/n-tlog n+O(1).

On the Existence of Nonadditive Quantum Codes

Most of the quantum error-correcting codes studied so far fall under the category of additive (or stabilizer) quantum codes, which are closely related to classical linear codes. The existence and general constructions of e_cient quantum codes that do not have such an underlying structure have remained elusive. Recently, specific examples of nonadditive quantum codes with minimum distance 2 have been presented. We, however, show that there exist infinitely many non-trivial nonadditive codes with different minimum distances, and high rates. In fact, we show that nonadditive codes that correct t errors can reach the asymptotic rate R = 1 - 2H2(2t/n), where H2(x) is the binary entropy function. In the process, we also develop a general set of sufficient conditions for a quantum code to be nonadditive. Finally, we introduce the notion of strongly nonadditive codes, and provide a construction for an ((11, 2, 3)) strongly nonadditive code.

Bounds for the weight distribution of weakly self-dual codes

Upper bounds are given for the weight distribution of binary weakly self-dual codes. To get these new bounds, we introduce a novel method of utilizing unitary operations on Hilbert spaces. This method is motivated by recent progress on quantum computing. This new approach leads to much simpler proofs for such genre of bounds on the weight distributions of certain classes of codes. Moreover, in some cases, our bounds are improvements on the earlier bounds. These improvements are achieved either by extending the range of the weights over which the bounds apply or by extending the class of codes subjected to these bounds.

Quantum Formulas: A Lower Bound and Simulation

We show that Nechiporuks method [I. Wegener, The Complexity of Boolean Functions, Teubner-Wiley, New York, 1987] for proving lower bounds for Boolean formulas can be extended to the quantum case. This leads to an

Partial recovery of entanglement in bipartite-entanglement transformations

\Omega(n^2/\log^2 n)

On Universal and Fault-Tolerant Quantum Computing

lower bound for quantum formulas computing an explicit function. The only known previous explicit lower bound for quantum formulas [A. Yao, Proceedings of 34th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1993, pp. 352--361] states that the majority function does not have a linear-size quantum formula. We also show that quantum formulas can be simulated by Boolean circuits of almost the same size.

Nanoelectronic implementations of reversible and quantum logic

Any deterministic bipartite-entanglement transformation involving finite copies of pure states and carried out using local operations and classical communication (LOCC) results in a net loss of entanglement. We show that for almost all such transformations, partial recovery of lost entanglement is achievable by using