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Physical Review A

Decoherence in large NMR quantum registers

Hans Georg Krojanski, Dieter Suter

Published Dec 26, 2006• 41 citations
Publisher

Abstract

Decoherence causes the decay of the quantum information that is stored in highly correlated states during quantum computation. It is thus a limiting factor for all implementations of a quantum computer. Because a scalable quantum computer with hundreds or thousands of qubits is not available yet, experimental data about decoherence rates was restricted to small quantum registers. With solid state nuclear magnetic resonance we create highly correlated multiqubit states that serve as a model quantum register and measure their decay. By measuring the decay as a function of the system size, we determined the scaling of the decoherence rate with the number of qubits. Using the same system, we also used decoupling techniques to reduce the coupling between system and environment and thereby the decoherence rate by more than an order of magnitude, independent of the system size. For the free decay as well as for the decoupled case, we found a relatively weak scaling with system size, which could be fitted to a power law $\ensuremath{\propto}{\overline{K}}^{p}$ with an exponent $p\ensuremath{\approx}1∕2$. This raises the prospect for large-scale quantum computation.

DOI
10.1103/physreva.74.062319

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Decoherence causes the decay of the quantum information that is stored in highly correlated states during quantum computation. It is thus a limiting factor for all implementations of a quantum computer. Because a scalable quantum computer with hundreds or thousands of qubits is not available yet, experimental data about decoherence rates was restricted to small quantum registers. With solid state nuclear magnetic resonance we create highly correlated multiqubit states that serve as a model quantum register and measure their decay. By measuring the decay as a function of the system size, we determined the scaling of the decoherence rate with the number of qubits. Using the same system, we also used decoupling techniques to reduce the coupling between system and environment and thereby the decoherence rate by more than an order of magnitude, independent of the system size. For the free decay as well as for the decoupled case, we found a relatively weak scaling with system size, which could be fitted to a power law \ensuremath∝K‾p\ensuremath{\propto}{\overline{K}}^{p}\ensuremath∝Kp with an exponent p\ensuremath≈1∕2p\ensuremath{\approx}1∕2p\ensuremath≈1∕2. This raises the prospect for large-scale quantum computation.
Among the most important parameters for the usefulness of quantum computers are the size of the quantum register and the decoherence time for the quantum information. The decoherence time is expected to get shorter with the number of correlated qubits, but experimental data are only available for small numbers of qubits. Solid-state nuclear magnetic resonance allows one to correlate large numbers of qubits (several hundred) and measure their decoherence rates. We use a modified magnetic dipole-dipole interaction to correlate the proton spins in a solid sample and observe the decay of the resulting highly correlated states. By systematically varying the number of correlated spins, we measure the increase of the decoherence rate with the size of the quantum register.
Among the biggest obstacles for building larger (and thus more powerful) quantum-information processors is decoherence, the decay of quantum-information by the coupling between the quantum register and its environment. Procedures for reducing decoherence processes will be essential for successful operation of larger quantum processors. We study model quantum registers consisting of up to 4900 qubits and measure their decay as a function of the register size. We demonstrate that appropriate sequences of qubit rotations reduce the coupling between system and environment for all sizes of the quantum register, thus preserving the quantum-information 50 times longer than without decoupling.
Effective quantum-information processing requires coherent control of large numbers of qubits on a time scale that is short compared to the decoherence time of the system. It is therefore important to extrapolate and measure decoherence times for large quantum registers and to determine the effect of different couplings between system and environment on the decoherence rate. For this purpose, we have experimentally realized a system that allows one to generate model quantum registers with more than 100 qubits and measure the decay of the information in these states while adjusting the strength of the interaction between the quantum register and the environment. Our results indicate a power-law dependence of the decoherence rate on the number of qubits in the system, with an exponent of the order of 0.5. This behavior remains qualitatively unchanged when the coupling strength to the environment is reduced by about an order of magnitude.
Nuclear magnetic resonance offers an appealing prospect for implementation of quantum computers, because of the long coherence times associated with nuclear spins, and extensive laboratory experience in manipulating the spins with radio frequency pulses. Existing proposals, however, suffer from a signal-to-noise ratio that decays exponentially in the number of qubits in the quantum computer. This places a severe limit on the size of the computations that can be performed by such a computer; estimates of that limit are well within the range in which a conventional computer taking exponentially more steps would still be practical. We give an NMR implementation in which the signal-to-noise ratio depends only on features of NMR technology, not the size of the computer. This provides a means for NMR computation techniques to scale to sizes at which the exponential speedup enables quantum computation to solve problems beyond the capabilities of classical computers.
The dynamical evolution of a quantum register of arbitrary length coupled to an environment of arbitrary coherence length is predicted within a relevant model of decoherence. The results are reported for quantum bits (qubits) coupling individually to different environments (``independent decoherence'') and qubits interacting collectively with the same reservoir (``collective decoherence''). In both cases, explicit decoherence functions are derived for any number of qubits. The decay of the coherences of the register is shown to strongly depend on the input states: We show that this sensitivity is a characteristic of both types of coupling (collective and independent) and not only of the collective coupling, as has been reported previously. A nontrivial behavior (``recoherence'') is found in the decay of the off-diagonal elements of the reduced density matrix in the specific situation of independent decoherence. Our results lead to the identification of decoherence-free states in the collective decoherence limit. These states belong to subspaces of the system's Hilbert space that do not get entangled with the environment, making them ideal elements for the engineering of ``noiseless'' quantum codes. We also discuss the relations between decoherence of the quantum register and computational complexity based on the dynamical results obtained for the register density matrix.
We investigate the decoherence process for a quantum register composed of NNN qubits coupled to an environment. We consider an environment composed of one common phonon bath and several electronic baths. This environment is relevant to the implementation of a charge based solid-state quantum computer. We explicitly compute the time evolution of all off-diagonal terms of the register's reduced density matrix. We find that in realistic configurations, ``superdecoherence'' and ``decoherence-free subspaces'' do not exist for an NNN-qubit system. This means that all off-diagonal terms decay, but not faster than ${e}^{\ensuremath{-}q(t)N}$, where q(t)q(t)q(t) is of the same order as the decay function of a single qubit.