Fast-forwarding Quantum Simulations

By Cristina Cirstoiu, Zoe Holmes, Ben Commeau, Marco Cerezo, Lukasz Cincio, Patrick Coles, and Andrew Sornborger

Simulations of physical systems are so computationally expensive that they comprise most of the global use of supercomputing power. But quantum computers, once realized, may provide an exponentially more efficient means of simulating quantum mechanical systems. This outcome could lead to improved pharmaceutical development, higher-temperature superconductors for more effective power transmission, new catalysts for nitrogen fixation and carbon capture, and significant research advancements in areas such as particle physics and chemistry. The fast forwarding algorithms that we are developing at Los Alamos National Laboratory will allow us to overcome limits that stifle real-world simulations.

The last few years have seen significant advances in quantum technology. The performance of small-scale quantum computers—based on various physical architectures—is constantly improving. Since these devices operate at a quantum scale, unwanted “noisy” interactions with the environment or within the devices themselves often hamper precise control and manipulation. While methods do exist to correct the quantum errors that occur throughout the computation, they require a large overhead in the number of qubits and a low noise threshold. These capabilities are beyond current and near-future quantum processors, which have been coined as noisy intermediate-scale quantum (NISQ) devices.

In the NISQ regime, the coherence time is the limited window for performing a quantum computation before the information is destroyed by noise, rendering the result meaningless. A new quantum algorithm called the variational fast forwarding algorithm (VFF) aims to overcome this noise-imposed barrier and enable long-time simulations of physical systems on NISQ devices.

The goal of VFF is to approximately diagonalize the short-time unitary evolution of a given quantum system — that is, to find a set of optimal angles that determine a structured, parameterized quantum circuit that approximates the short-time evolution. One can then use this approximate diagonalization to simulate longer evolutions simply by modifying the angles “by hand.” Consequently, VFF allows quantum simulations to fast-forward; this means that longer simulations are implemented within a fixed computational time. This technique is in contrast to many other methods—such as the iterated Trotter approach—that require repeated application of a short-time unitary and thus a number of quantum computational time-steps that scales with the length of time being simulated.

The concept of Variational Fast Forwarding (VFF). (a) A Trotterization-based quantum simulation with \(N=5\) timesteps. This simulation runs past the coherence limit of the quantum architecture. (b) A VFF-based quantum simulation. An approximate diagonalization of a short-time simulation is found variationally. Using the eigenvector \(W\) and diagonal \(D\) unitaries that were learned, an arbitrary length simulation is implemented by modifying the parameters in \(D\). As long as VFF results in few enough gates that the circuit does not exceed the coherence time, longer simulations can be performed than the standard method in (a). This figure is used with permission from SpringerNature under the Creative Commons Attribution v4.0 International license. It was originally published in "Variational fast forwarding for quantum simulation beyond the coherence time" by Cristina Cirstoiu, Zoe Holmes, Joseph Iosue, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger, npj Quantum Information 6, no. 1 (2020): 1-10.

One can find the diagonalization at the heart of VFF via a hybrid quantum-classical variational approach. In this method, researchers compute only a small component of the algorithm on a quantum computer — typically a cost function that is difficult to calculate classically but relatively easy to calculate with quantum mechanics. The rest of the algorithm—usually an optimization loop—is run on a classical computer. This is clearly less demanding than running the full algorithm on the quantum processor and is therefore more suitable for NISQ devices. The cost function for VFF utilizes the Hilbert-Schmidt inner product between the short-time unitary evolution and a possible diagonal compilation of that unitary. Researchers can minimize this cost using machine learning techniques, such as gradient descent, to approximately diagonalize a system’s short-time evolution.

We have implemented VFF on quantum hardware for toy examples with promising results. As the total quantum simulation times increase, the accuracy of a single-qubit simulation drops more slowly with VFF than with other leading methods, such as the iterated Trotter approach. This result demonstrates that VFF can indeed perform quantum simulations beyond the coherence time of current quantum computers.

VFF is fundamentally limited by an error in the initial approximation for the system’s short-time evolution. Because one typically approximates the short-time evolution with a Trotter approximation, this error carries over to the final simulation. A second method that we have developed—called the variational Hamiltonian diagonalization (VHD) algorithm—avoids this error by directly diagonalizing the Hamiltonian of the system of interest instead of an approximation of the system’s short-time evolution. We construct VHD's cost function from the Hilbert-Schmidt distance between the quantum simulation’s Hamiltonian and a parameterized diagonal version, and evaluate it on a quantum computer by expanding the Hamiltonians as a weighted sum of unitaries. In addition to removing the Trotter error of VFF, the algorithm results in a shorter circuit; however, this increases the number of quantum circuit evaluations. Therefore, VHD is expected to come into its own once quantum computing power becomes more accessible.

VFF and VHD introduce the possibility of performing useful quantum simulations on near-term hardware. It will be intriguing to see the impact of these algorithms on basic science, technology, and beyond.

Cristina Cirstoiu is a research scientist at Cambridge Quantum Computing. She was previously a postdoctoral researcher at Oxford University and holds a Ph.D. in quantum information theory from Imperial College London. Zoe Holmes is a postdoctoral researcher at Los Alamos National Laboratory (LANL). She holds a Ph.D. in quantum information theory from Imperial College London. Benjamin Commeau recently graduated from the University of Connecticut with a Ph.D. in condensed matter and quantum computation. He worked with LANL’s Quantum Computing group during a research fellowship. Marco Cerezo is a postdoctoral research at LANL. He holds a Ph.D. in physics from the National University of La Plata in Buenos Aires, Argentina. Lukasz Cincio, Patrick Coles, and Andrew Sornborger are all staff researchers at LANL.