Recent theoretical breakthroughs have settled two long-standing questions
about the viability of simulating quantum systems on future quantum
computers, overcoming challenges from complexity analyses to enable more
advanced algorithms. Featured in two publications, the work by a quantum
team at Los Alamos National Laboratory shows that physical properties of
quantum systems allow for faster simulation techniques.

"Algorithms based on this work will be needed for the first full-scale
demonstration of quantum simulations on quantum computers," said Rolando
Somma, a quantum theorist at Los Alamos and coauthor on the two papers.

### Low-energy quantum states key to faster quantum simulation

The paper "Hamiltonian simulation in the low-energy subspace" demonstrates
that the complexity of a quantum simulation algorithm depends on the
relevant energy scale and not the full range of energies of the system, as
previously thought. In fact, some quantum systems can have states of
unbounded energies, hence simulations would prove intractable even on large
quantum computers.

This new research found that, if a quantum system explores the low-energy
states only, it could be simulated with low complexity on a quantum computer
without errors crashing the simulation.

"Our work provides a path to a systematic study of quantum simulations at
low energies, which will be required to push quantum simulations closer to
reality," said Burak Şahinoğlu, a theoretical physicist at Los Alamos and
lead author on the paper, published in the journal Quantum Information, a
Nature partner journal.

"We show that at every step of the algorithm, you never escape to the very
large energies," said Somma. "There's a way of writing your quantum
algorithm so that after each step you're still within your low-energy
subspace."

The authors said their research applies to a large class of quantum systems
and will be useful in simulating quantum field theories, which describe
physical phenomena within their low-energy states.

### Fast-forwarding of quantum systems bypasses the time-energy uncertainty principle

The other paper, "Fast-forwarding quantum evolution," a collaboration with
Caltech's Shouzhen Gu—a former Los Alamos quantum computing summer school
student—is published in Quantum. It shows three quantum systems in which a
quantum simulation algorithm can run faster—and in some cases exponentially
faster—than the limits suggested by the time-energy uncertainty
principle.

"In quantum mechanics, the best precision that can be achieved when
measuring a system's energy scales, in general, with the inverse of the
duration of the measurement," said Somma.

"However, this principle does not apply to all quantum systems, especially
those that have certain physical features," said Şahinoğlu.

The authors showed that when this principle is bypassed, such quantum
systems can also be simulated very efficiently, or fast-forwarded, on
quantum computers.

## References:

Burak Şahinoğlu et al, Hamiltonian simulation in the low-energy subspace,
npj Quantum Information (2021).
DOI: 10.1038/s41534-021-00451-w

Shouzhen Gu et al, Fast-forwarding quantum evolution, Quantum (2021).
DOI: 10.22331/q-2021-11-15-577

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