What's next for applied quantum computing?

NISQ (noisy intermediate-scale quantum) algorithms generated a lot of excitement and a lot of publications - the 2022 review has amassed almost 2000 citations! Nowadays the tone is more subdued, with many experts believing any useful practical applications of quantum processors will need quantum error correction. The new hot topics are understanding how to make useful error correction a reality, and what might be done with a few hundred logical qubits. 

What then should a new student interested in applied quantum computing focus on?

Ryan Babbush and collaborators already argued in 2021 that algorithms with quadratic speedups won't be useful in practice. So sorry, but we won't be able to solve complex industry optimization problems using Grover search. However, their analysis indicated that quartic speedups and beyond could be practically useful. Which quantum algorithms have this property?

Consulting the excellent review article Quantum algorithms: A survey of applications and end-to-end complexities, there are only a few examples of known or suspected quartic or beyond end-to-end quantum speedups! They are:

Tensor principal component analysis (PCA). Ordinary PCA is a data reduction step widely used in data analysis and machine learning. It's not yet clear what tensor PCA might be useful for, but if an application can be found quantum computers will probably give a useful speedup.

Topological data analysis (TDA). This is another promising direction where a useful speedup for certain problems is possible. Following an initial buzz of excitement in 2022, it's unclear whether there are practical applications for where such a speedup can be useful. Recently-developed quantum-inspired classical algorithms will be useful to identify potential use-cases for quantum TDA.

On the classical computing side, quantum-inspired tensor network methods are very promising for near-term applications.  

There are also other approaches (QAOA, quantum machine learning) which attracted a lot of interest since 2020 and are still being explored theoretically, but at least in their present formulations they seem unable to provide a useful speedup for classical problems, with their most promising applications related to directly studying or simulating certain quantum systems. Thus, interest has shifted from "beating" classical methods on carefully-selected problems to better understanding the foundations of quantum machine learning. While this is a fascinating topic, it is at this stage it is more theoretical than applied research.

arXiv highlights

 A randomized benchmarking suite for mid-circuit measurements

Mid-circuit measurements of ancilla qubits form a key component of near-term quantum algorithms including NISQ-TDA, future quantum error correction techniques, and quantum algorithms for fault-tolerant quantum computers. This preprint from IBM presents techniques for quantifying the performance and error rates of mid-circuit measurements which are now supported by IBM quantum processors.

 

Laser-guided lightning

 

Focused high-intensity laser pulses can ionize air molecules, creating channels of air with higher electrical conductivity which can be used to guide and induce lightning strikes. Peaceful applications include protecting airports and rockets from lightning strikes. No doubt there are also military applications of this kind of research.

 

Large-Scale Simulation of Quantum Computational Chemistry on a New Sunway Supercomputer

 

Quantum chemistry calculations using classical supercomputers. Exponential memory scaling is avoided by using density matrix embedding theory to eliminate irrelevant degrees of freedom and matrix product states to efficiently capture quantum correlations. Apparently this approach can handle chemical systems with up to 100 atoms and it may also be useful for benchmarking variational quantum algorithms run on near-term quantum processors.

 

The Quantum Approximate Optimization Algorithm performance with low entanglement and high circuit depth

 

"Our simulation with the lowest bond dimension D = 2—for which the fidelity is close to zero—obtains a solution to the classical problem providing the depth of the circuit is high enough p ≈ 20, even with a sub-optimal choice of parameters and single deterministic sampling of bitstrings...In conclusion, we observe that entanglement plays a minor role in finding the solution to the classical problems studied here for large-depth QAOA." More evidence that your "quantum" optimization algorithm can be efficiently run on an ordinary classical computer.

Computing Electronic Correlation Energies using Linear Depth Quantum Circuits

 

Finally, I am advertising our own recent work on incorporating electronic correlation effects into shallow quantum circuits suitable for near-term noisy quantum processors. 

Thermofield double states

Quantum simulation of materials at finite temperatures requires the generation of thermal quantum states.

Thermal quantum states are given by density matrices in which the eigenstate occupation probability follows the Boltzmann distribution.

Such density matrices cannot be generated from pure unitary quantum evolution. They either require the quantum system to interact with some environment, or to trace over some components of an entangled quantum state.

The latter approach is the most promising for the generation of thermal states of arbitrary quantum Hamiltonians. One approach based on thermofield double states, enables the generation of an N qubit thermal state using a 2N qubit pure state, i.e. two copies of the system of interest.

How to prepare thermofield double states?

One approach recently implemented in ion trap and superconducting qubit experiments employs the quantum approximate optimization algorithm (QAOA). The idea is that the infinite temperature (fully mixed) state is easy to prepare and the ground state of a simple (mixing) Hamiltonian that entangles pairs of qubits, one from the system of interest, and the other from the subsystem to be traced out. One can also identify a Hamiltonian that whose ground state describes the system of interest at zero temperature.

In the limit of a large number of steps, QAOA effectively performs an adiabatic transformation from the infinite temperature double state (easy to prepare) to the zero temperature one (hard to prepare). This suggests it should also be able to well-approximate intermediate temperature states using comparatively shallow circuits by solving a variational optimization problem; the cost function used is some measure of fidelity of the obtained density matrix with respect to a thermal state at the desired temperature.

Other approaches are based on deterministic algorithms, but require deeper circuits not really suitable for near-term quantum processors, for example relying on the phase estimation algorithm to obtain the desired thermal state.

Last year researchers from China proposed an alternate scheme which uses a continuous variable quantum state (qmode) to assist with the preparation of the thermofield double state. In this case, the challenges appear to be preparation of the required qmode resource state, decomposing the target Hamiltonian into a series of unitaries controlled by the qmode, and finite precision in measuring the quadrature of the qmode.

Since future materials science applications of quantum processors will primarily be concerned with simulating finite temperature properties, developing more efficient and robust schemes for the preparation of thermal states will be a topic of growing interest in the coming years.