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Physical Review A

Quantum and classical parallelism in parity algorithms for ensemble quantum computers

Ralf Stadelhofer, Dieter Suter, Wolfgang Banzhaf

Published Mar 28, 2005• 20 citations
Publisher

Abstract

The determination of the parity of a string of $N$ binary digits is a well-known problem in classical as well as quantum information processing, which can be formulated as an oracle problem. It has been established that quantum algorithms require at least $N∕2$ oracle calls. We present an algorithm that reaches this lower bound and is also optimal in terms of additional gate operations required. We discuss its application to pure and mixed states. Since it can be applied directly to thermal states, it does not suffer from signal loss associated with pseudo-pure-state preparation. For ensemble quantum computers, the number of oracle calls can be further reduced by a factor ${2}^{k}$, with $k∊{1,2,\dots{},{\mathrm{log}}_{2}(N∕2)}$, provided the signal-to-noise ratio is sufficiently high. This additional speed-up is linked to (classical) parallelism of the ensemble quantum computer. Experimental realizations are demonstrated on a liquid-state NMR quantum computer.

DOI
10.1103/physreva.71.032345

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The determination of the parity of a string of NNN binary digits is a well-known problem in classical as well as quantum information processing, which can be formulated as an oracle problem. It has been established that quantum algorithms require at least N∕2N∕2N∕2 oracle calls. We present an algorithm that reaches this lower bound and is also optimal in terms of additional gate operations required. We discuss its application to pure and mixed states. Since it can be applied directly to thermal states, it does not suffer from signal loss associated with pseudo-pure-state preparation. For ensemble quantum computers, the number of oracle calls can be further reduced by a factor 2k{2}^{k}2k, with k∊1,2,…,log2(N∕2)k∊{1,2,\dots{},{\mathrm{log}}_{2}(N∕2)}k∊1,2,…,log2​(N∕2), provided the signal-to-noise ratio is sufficiently high. This additional speed-up is linked to (classical) parallelism of the ensemble quantum computer. Experimental realizations are demonstrated on a liquid-state NMR quantum computer.
Quantum algorithms are known for providing more efficient solutions to certain computational tasks than any corresponding classical algorithm. Here we show that a single qudit is sufficient to implement an oracle based quantum algorithm, which can solve a black-box problem faster than any classical algorithm. For 2d permutation functions defined on a set of d elements, deciding whether a given permutation is even or odd, requires evaluation of the function for at least two elements. We demonstrate that a quantum circuit with a single qudit can determine the parity of the permutation with only one evaluation of the function. Our algorithm provides an example for quantum computation without entanglement since it makes use of the pure state of a qudit. We also present an experimental realization of the proposed quantum algorithm with a quadrupolar nuclear magnetic resonance using a single four-level quantum system, i.e., a ququart.
Quantum algorithms are known for providing more efficient solutions to certain computational tasks than any corresponding classical algorithm. Here we show that a single qudit is sufficient to implement an oracle based quantum algorithm, which can solve a black-box problem faster than any classical algorithm. For 2d permutation functions defined on a set of d elements, deciding whether a given permutation is even or odd, requires evaluation of the function for at least two elements. We demonstrate that a quantum circuit with a single qudit can determine the parity of the permutation with only one evaluation of the function. Our algorithm provides an example for quantum computation without entanglement since it makes use of the pure state of a qudit. We also present an experimental realization of the proposed quantum algorithm with a quadrupolar nuclear magnetic resonance using a single four-level quantum system, i.e., a ququart.
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Consider a function fff which is defined on the integers from 1 to NNN and takes the values $\ensuremath{-}1$ and +1+1+1. The parity of fff is the product over all xxx from 1 to NNN of f(x)f(x)f(x). With no further information about fff, to classically determine the parity of fff requires NNN calls of the function fff. We show that any quantum algorithm capable of determining the parity of fff contains at least N/2N/2N/2 applications of the unitary operator which evaluates fff. Thus, for this problem, quantum computers cannot outperform classical computers.
The quantum search problem is an important problem due to the fact that a general NP problem can be solved efficiently by an unsorted quantum search algorithm. Here it has been shown that the quantum search problem could be solved in polynomial time on an NMR quantum computer. The NMR ensemble quantum computation is based on the quantum mechanical unitary dynamics that both a closed quantum system and its ensemble obey the same quantum mechanical unitary dynamics instead of on the pseudopure state or the effective pure state of the classical NMR quantum computation. Based on the new principle the conventional NMR multiple-quantum spectroscopy has been developed to solve experimentally the search problem. The solution information of the search problem is first loaded on the unitary evolution propagator which is constructed with the oracle unitary operation and oracle-independent unitary operations and is used to excite the multiple-quantum coherence in a spin ensemble. Then the multiple- quantum spectroscopy is used to extract experimentally the solution information. It has been discussed how to enhance the output NMR signal of the quantum search NMR multiple-quantum pulse sequence and some approaches to enhancing the NMR signal are also proposed. The present work could be helpful for conventional high field NMR machines to solve efficiently the quantum search problem.