Double-Bracket Iterations: A New Pathway for Quantum Computing Algorithms

Double-bracket iterations, a proposed framework for obtaining diagonalizing quantum circuits, could advance quantum computing by approximating eigenstates of relevant quantum models with few recursion steps. This method, which interlaces evolutions generated by the input Hamiltonian with diagonal evolutions, does not require any qubit overheads or controlled-unitary operations. It is more suitable for near-term quantum computing experiments due to its lower implementation cost compared to quantum phase estimation. The Głazek-Wilson-Wegner (GWW) flow, a type of double-bracket flow, can compile diagonalizing quantum circuits, providing a feasible solution for quantum computing.

What are Double-Bracket Iterations in Quantum Computing?

Double-bracket iterations are a proposed framework for obtaining diagonalizing quantum circuits. These circuits are implemented on a quantum computer by interlacing evolutions generated by the input Hamiltonian with diagonal evolutions, which can be chosen variationally. This method does not require any qubit overheads or controlled-unitary operations, but it is recursive, which means the circuit depth grows exponentially with the number of recursion steps.

To make near-term implementations viable, the proposal includes optimization of diagonal evolution generators and of recursion step durations. Numerical examples show that the expressive power of double-bracket iterations suffices to approximate eigenstates of relevant quantum models with few recursion steps. Compared to brute-force optimization of unstructured circuits, double-bracket iterations do not suffer from the same trainability limitations.

Moreover, with an implementation cost lower than required for quantum phase estimation, they are more suitable for near-term quantum computing experiments. This work opens a pathway for constructing purposeful quantum algorithms based on so-called double-bracket flows, also for tasks different from diagonalization, and thus enlarges the quantum computing toolkit geared towards practical physics problems.

How Can Double-Bracket Iterations Advance Quantum Computing?

Studying quantum many-body systems is an area where quantum computing may lead to practical advances outside the scope of what we can compute numerically. We often gain physics understanding by analyzing eigenstates and eigenvalues of quantum models, so quantum algorithms for diagonalization could be key for achieving new insights.

To approximate eigenstates on a quantum computer, a basic idea is to variationally find appropriate parameters for a sequence of quantum gates. However, the optimization needed for that is made difficult by a phenomenon referred to as barren plateaus. Unstructured ansatzae, e.g., circuits composed of CNOT gates and single-qubit rotations, have been studied extensively, but there are very few exceptional cases when it has been possible to find parametrizations that generalize to system sizes larger than a handful of qubits.

Recently, a more algorithmic approach based on the classical Lanczos algorithm, widely employed for numerical diagonalization, has found application in a quantum device. Its innovation consists in implementing imaginary-time evolutions using accessible unitary operations.

What is the Role of Głazek-Wilson-Wegner (GWW) Flow in Quantum Computing?

The Głazek-Wilson-Wegner (GWW) flow can play a conceptual role for quantum computing as it provides a feasible solution to the task of compiling diagonalizing quantum circuits. Applications of GWW flow in condensed-matter physics have been explored using classical computers. The GWW flow is an example of nonlinear differential equations called double-bracket flows, whose mathematics is covered in the monograph by Helmke and Moore.

With near-term quantum computing applications in mind, we will depart from the GWW flow. Instead, we will consider double-bracket iterations, which can recover continuous flows as a special limit but give flexibility to attempt lowering implementation cost. It appears difficult to characterize the precise efficacy of variational double-bracket iterations analytically, but numerical simulations show that just a handful of recursion steps can yield surprisingly good approximations of low-energy eigenstates of the quantum Ising model.

How are Double-Bracket Iterations Implemented in Quantum Computing?

The recursive character of double-bracket iterations leads to an exponential runtime of the quantum algorithms in the number of iteration steps. Numerical exploration demonstrates that few steps of a diagonalization double-bracket iteration suffice to achieve relevant state preparations. Double-bracket iterations can be approximated by quantum circuits using Hamiltonian simulation. Simulating evolution under input Hamiltonians is being actively explored in experiments, and so the proposed quantum algorithm lends itself towards near-term experiments. In particular, no controlled-unitary operations are needed.

What are the Future Applications of Double-Bracket Iterations?

Continuous double-bracket flows, in particular, a variant of a double-bracket iteration is proposed which converges to the GWW flow. The obtained runtime is not efficient, but the presented theory should be relevant for anyone wishing to explore quantum computing applications of double-bracket flows for tasks other than diagonalization.

Double-bracket iterations are discussed in the context of other approaches to constructing quantum algorithms. In particular, quantum dynamic programming can reduce the circuit depth of recursive double-bracket iterations, thus potentially making them more efficient and applicable in a wider range of quantum computing tasks.