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Physical Review Letters

Scaling of Decoherence in Wide NMR Quantum Registers

Hans Georg Krojanski, Dieter Suter

Published Aug 24, 2004• 135 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.93.090501

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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.
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.
Abstract In the past decade, nuclear magnetic resonance (NMR) has been actively used to study the basic principles of quantum computers. It is assumed that quantum correlations play a significant role in their performance. They exist at both low and high temperatures. At the same time, the time correlation functions of nuclear spin systems of solids determine the observed signals in traditional NMR implementations. The separation of such signals into quantum and classical components has not previously been carried out and will be performed in this study for the most important of the correlational functions observed in magnetic resonance: the free induction decay (FID).
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.
We present an improved approach for the study of the evolution of spin correlations and decoherence in multiple quantum nuclear magnetic resonance experiments. The infinite system, constituted by the protons of a polycrystalline adamantane sample, evolves under a double quantum Hamiltonian. The distribution of multiple quantum coherence orders is represented by a contribution of spin clusters with different sizes that exchange spins, increasing their size with the evolution time. A cluster with nearly exponential growth at all times is observed, in agreement with previous models. Remarkably, a small cluster that stabilizes in a size corresponding to 18 correlated spins is revealed. In addition, by performing a renormalization of the obtained data with the experimental Loschmidt echo, the contribution of the different clusters to the observable signal is determined. This procedure accounts for the effect of decoherence on the evolution of the system, and allows setting the range of confidence of the experimental data. Our analysis confirms the natural hint that, correlated states involving higher coherence orders are far more sensitive to the uncontrolled decoherent interactions, than those involving lower orders.
We survey results on the decay of multiqubit entanglement due to internal interactions between qubits. Dipole-dipole interaction induces decoherence of strongly entangled nuclear spins. The dynamics of spin clusters can be described as quantum decoherence due to an effective composite bath consisting of fully correlated and uncorrelated parts. The rate of decoherence scales up as a square root of the number of entangled spins, resulting in linear scaling of a measure of quantum noise. Our theory is consistent with a recent experiment.
Nuclear magnetic resonance (NMR) provides an experimental setting to explore physical implementations of quantum information processing (QIP). Here we introduce the basic background for understanding applications of NMR to QIP and explain their current successes, limitations and potential. NMR spectroscopy is well known for its wealth of diverse coherent manipulations of spin dynamics. Ideas and instrumentation from liquid state NMR spectroscopy have been used to experiment with QIP. This approach has carried the field to a complexity of about 10 qubits, a small number for quantum computation but large enough for observing and better understanding the complexity of the quantum world. While liquid state NMR is the only present-day technology about to reach this number of qubits, further increases in complexity will require new methods. We sketch one direction leading towards a scalable quantum computer using spin 1/2 particles. The next step of which is a solid state NMR-based QIP capable of reaching 10-30 qubits.