# Carving Up Infinite Quantum Spaces into Simpler Surrogates

March 7, 2024Researchers at JQI and the Joint Center for Quantum Information and Computer Science (QuICS) have constructed a new mathematical shortcut for a wide range of calculations involving quantum systems. Their results, which they reported on Feb. 8, 2024, in the journal *Physical Review X*, answered an open question about whether such a shortcut even existed, and the authors say that their work could lead to more efficient benchmarking for quantum devices, as well as more efficient ways of representing quantum states on classical hardware.

The new shortcut, known as a continuous variable (CV) quantum state design, expands previously known tools to a new realm—that of CV quantum systems, in which quantum measurements can yield an infinite number (i.e. a continuum) of possible values. This is in contrast to measurements of qubits, which have a finite set of discrete outcomes (just two for a single qubit measurement). Many promising platforms for quantum information science and its applications are CV systems—including optical and microwave cavities, optical fibers and nanoacoustic resonators—to which the earlier shortcuts don’t apply.

In essence, a quantum state design approximates the features of an infinite set of quantum states using a finite set. This finite set is a kind of wireframe representation of the infinite set, and it acts like a perfect stand-in in certain calculations. The upshot is that quantum calculations that ordinarily require averaging over the infinite set (using a mathematical integral) turn into simpler sums over the finite set.

The existence of CV state designs had been in question for years. Earlier work showed that the most straightforward attempts to find CV state designs were doomed to fail—a fact conveyed during a conference presentation that one of the authors attended in 2011. “That talk was tantalizing but intimidating because of the math behind it,” says QuICS Fellow Victor Albert, who is also a physicist at the National Institute of Standards and Technology. “It left open the question of whether CV designs exist at all.”

At the time, non-CV quantum state designs were all the rage, and researchers were looking for natural extensions to CV systems. In particular, the work showed that CV designs couldn’t be constructed out of Gaussian states, which are the CV analog of the kinds of states used in non-CV designs. If Gaussian states wouldn’t work, Albert assumed that some other set of states would suffice. When he returned to the problem in 2021, he assigned it to Joe Iosue, a graduate student in physics at the University of Maryland who is a member of JQI and QuICS.

“The first thing to do was just trying to create a design,” says Iosue, who is the first author of the new paper. “After we tried for a while, we thought ‘Jeez, this is really hard. Maybe it’s not possible.’ So we tried to prove it’s not possible.”

Iosue started building the case against CV state designs, aiming to finally close the door on the entire avenue of research. He dug through old textbooks for inspiration and recast the problem from the language of linear algebra (useful for non-CV state designs) to measure theory—the branch of mathematics that generalizes the idea of measuring length, area and volume beyond simple geometrical shapes.

“Ultimately what unlocked it was I went through literally a dozen measure theory textbooks looking for theorems that would be useful,” Iosue says.

He eventually found the key theorem in an old book and succeeded in proving that, in general, CV quantum state designs couldn’t exist. But at the same time, he and his colleagues noticed that their proof made an assumption that need not hold. If they relaxed that assumption, they could construct CV state designs, but the cost would be that the states themselves would become unphysical—to create a CV quantum state design, they had to include at least some states with infinite energy.

This would normally be a problem, since a state with infinite energy could never be prepared or used in a lab. But if you turn that set of states into a set of measurements, it turns out that you can make those measurements in a lab—at least theoretically.

Practically, it still might take a while before experiments can make use of the construction, says Albert. “We’re pretty far away because the states we’ve identified are not very close to the ‘easy’ type of Gaussian states available to everybody,” he says. “It’s a challenge measuring in the basis of these states.”

But assuming researchers can overcome the technical challenge of making these measurements, the authors point to some straightforward applications.

“The main usefulness in terms of quantum computing is to benchmark stuff,” says Kunal Sharma, a former QuICS Hartree Postdoctoral Fellow and another author of the new paper. “Suppose you want to implement a gate. How do you benchmark its quality?”

Benchmarking is a common framework for determining whether a quantum device is functioning as expected. It compares the output of a real quantum device (fed with random inputs) to the expected output of an ideal device, and that comparison boils down to computing a number that characterizes the quality of the quantum process. Often the comparison involves the average value of a particular set of quantum measurements, and the average is taken over all possible quantum states—a natural place for CV designs to improve efficiency.

In addition to reducing the overhead for benchmarking quantum devices, another area that state designs have been useful in the past is in shadow tomography. Shadows are a way to characterize a quantum state using a reduced set of parameters—like rebuilding the image of something from the shadow it casts on the wall. CV state designs provide a more efficient way of representing arbitrary quantum states, allowing for more efficient computations on the state itself—even on non-quantum hardware.

In related work that has been accepted to *PRX Quantum*, two of the authors of the CV state design paper characterized these shadow methods for estimating CV quantum states, albeit in a setting outside of strict CV state designs. They proved worst-case bounds on the number of measurements needed to generate estimates of sufficient quality, and they performed numerical experiments that showed shadow methods typically beat the worst-case bounds by a fair margin.

Albert says that continuing to study CV state designs and CV quantum information in general is a practical necessity, even if most quantum computing work remains couched in the finite language of qubits. “We have to use continuous systems whether we like it or not,” he says, “and that requires a different approach.”

*Story by Chris Cesare*

*In addition to Albert, Iosue and Sharma, this paper had one more author: QuICS Fellow Michael Gullans, who is also a physicist at NIST.*

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