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arXiv:quant-ph/0601021 · Physical Review Letters

Limitations of Quantum Simulation Examined by Simulating a Pairing Hamiltonian Using Nuclear Magnetic Resonance

Kenneth R. Brown, Robert Clark, Isaac L. Chuang

Published Aug 4, 2006• 98 citations
arXivHTMLPDF

Abstract

Quantum simulation uses a well-known quantum system to predict the behavior of another quantum system. Certain limitations in this technique arise, however, when applied to specific problems, as we demonstrate with a theoretical and experimental study of an algorithm proposed by Wu, Byrd, and Lidar [Phys. Rev. Lett. 89, 057904 (2002).10.1103/PhysRevLett.89.057904] to find the low-lying spectrum of a pairing Hamiltonian. While the number of elementary quantum gates required scales polynomially with the size of the system, it increases inversely to the desired error bound E. Making such simulations robust to decoherence using fault tolerance requires an additional factor of approximately 1/E gates. These constraints, along with the effects of control errors, are illustrated using a three qubit NMR system.

DOI
10.1103/physrevlett.97.050504

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Quantum simulation uses a well-known quantum system to predict the behavior of another quantum system. Certain limitations in this technique arise, however, when applied to specific problems, as we demonstrate with a theoretical and experimental study of an algorithm proposed by Wu, Byrd, and Lidar [Phys. Rev. Lett. 89, 057904 (2002).10.1103/PhysRevLett.89.057904] to find the low-lying spectrum of a pairing Hamiltonian. While the number of elementary quantum gates required scales polynomially with the size of the system, it increases inversely to the desired error bound E. Making such simulations robust to decoherence using fault tolerance requires an additional factor of approximately 1/E gates. These constraints, along with the effects of control errors, are illustrated using a three qubit NMR system.
The simulation of the spectra measured in nuclear magnetic resonance (NMR) spectroscopy experiments is a computationally non-trivial problem which, due to its natural interpretation as a quantum spin problem, maps in a straightforward way to a quantum computer. As such, it represents a problem for which such a device may provide some practical advantage over traditional computing methods. In order to understand the extent to which such problems may indeed provide examples of useful quantum advantage, it is important to understand the limitations of existing classical simulation methods. In this work, we benchmark our own classical solver designed to study such problems. This solver uses a clustering approximation to achieve a resource scaling which is linear in the total number of nuclear spins in a given molecule, for a fixed cluster size. The success of such an approximation would present a stark repudiation to the common claim that such problems require an exponential scaling of resources, the very claim which makes simulating an NMR spectra a candidate for quantum advantage. Our benchmarking results indicate that our approximation performs well throughout, and even somewhat beyond, the more typical experimental regimes. We discuss what implications this may have for future efforts to demonstrate quantum advantage in the context of NMR.
Advanced atomic magnetometers have made it possible to acquire nuclear magnetic resonance spectra in zero to ultralow magnetic fields. This regime carries the benefit of compact, low-cost instrumentation with reduced spin relaxation effects and the ability to probe phenomena that are inaccessible in conventional high-field experiments. A trade-off is that the resulting spectra must be interpreted using simulations that are taxing for classical computation. Working by example for small-molecule and protein spectroscopy, we demonstrate that these simulations are a promising target for fault-tolerant quantum computation. Our holistic analysis spans from input selection to the construction of explicit circuits for qubitized quantum dynamics. By maintaining parity with experimental requirements, we demonstrate how certain cases might be especially promising for early fault-tolerant architectures. Notably, valuable natural products and small proteins could be simulated with a few hundred logical qubits and less than 10 12 T gates, corresponding to days of runtime with foreseeable hardware based on spin or superconducting qubits.
With the latest advances in quantum computing technology, we are gradually moving from the noisy intermediate-scale quantum (NISQ) era characterized by hardware limited in the number of qubits and plagued with quantum noise, to the age of quantum utility where both the newest hardware and software methods allow for tackling problems which have been deemed difficult or intractable with conventional classical methods. One of these difficult problems is the simulation of one-dimensional (1D) nuclear magnetic resonance (NMR) spectra, a major tool to learn about the structure of molecules, helping the design of new materials or drugs. Using advanced error mitigation and error suppression techniques from Q-CTRL together with the latest commercially available superconducting-qubit quantum computer from IBM and trapped-ion quantum computer from IonQ, we present the quantum Hamiltonian simulation of liquid-state 1D NMR spectra in the high-field regime for spin systems up to 34 spins. Our pipeline has a major impact on the ability to execute deep quantum circuits with the reduction of quantum noise, improving mean square error by a factor of 22. It allows for the execution of deep quantum circuits and obtaining salient features of the 1D NMR spectra for both 16-spin and 22-spin systems, as well as a 34-spin system, which lies beyond the regime where unrestricted full Liouvillespace simulations are practical (32 spins, the Liouville limit). Our work is a step toward near-term quantum utility in NMR spectroscopy.
Quantum algorithms for problems such as integer factorization and database search have spurred great interest in quantum computation. However, the construction of scalable quantum computers has proved to be extremely difficult. Among the challenging requirements that must be fulfilled in the “standard model” of quantum computation, one must be able to initialize qubits into known pure states. In addition, quantum computers must be protected from noise due to unwanted interactions with their environment. This work is concerned with two theoretical issues motivated by these difficulties. First, I analyze the efficiency of an algorithm for an alternate model for quantum computation that is motivated by bulk nuclear magnetic resonance. In such ensemble systems, the initial state is not a fiducial pure state, but is instead a maximally-mixed density operator. The algorithm that I consider allows one to estimate the free energy of arbitrary spin-1/2 lattice models. To determine the efficiency of this algorithm, I calculate the computation time required to estimate the free energy to bounded error as a function of the lattice size. In the absence of stochastic fluctuations in the measurement output, it is found that the algorithm is efficient. However, evidence is presented that suggests that the algorithm becomes exponentially sensitive to fluctuations as the lattice size increases. While techniques such as quantum error-correcting codes and decoherence-free subsystem encodings have been devised to mitigate errors due to unwanted environmental couplings, these methods require many additional qubits or complicated encodings. Here, I investigate a simple approach to reduce errors in the quantum search algorithm due to a collective decoherence model. This method takes advantage of the freedom inherent in compiling the search algorithm into fundamental gates. Transition rate calculations and more rigorous quantum master equation simulations are carried out for small-qubit instances to contrast the performance of the original and modified algorithm. It is shown that the expected computational effort can be reduced by 22% for selected five-bit instances of the quantum search algorithm. While this approach does not constitute a general strategy for quantum error-correction, it illustrates the importance of judicious gate design in mitigating decoherence.
Recently developed quantum algorithms suggest that in principle, quantum computers can solve problems such as simulation of physical systems more efficiently than classical computers. Much remains to be done to implement these conceptual ideas into actual quantum computers. As a small-scale demonstration of their capability, we simulate a simple many-fermion problem, the Fano-Anderson model, using liquid-state nuclear magnetic resonance (NMR). We carefully designed our experiment so that the resource requirement would scale up polynomially with the size of the quantum system to be simulated. The experimental results obtained give us an insight to the quantum control required when simulating quantum systems with NMR techniques. The simulation of other physical systems, with different particle statistics, is also discussed.