# Quantum Volume

Quantum Volume (QV) is a metric that indicates the largest square quantum circuit that a quantum computer can successfully implement. The dimensions of the square are width, which is the number of qubits, and depth, which is the number of time steps. Scoring takes into consideration qubit counts, gate errors, measurement errors, crosstalk, and connectivity. The protocol mainly tests the execution of randomized two-qubit gates acting in parallel but may use other gates, as well. Higher QV scores indicate more potential computational power.

For more information, our article “Quantum Computer Technology: Architecture, Advantages and Disadvantages” provides an overview of quantum computing technology, with QV mentioned as a metric. Also, the Ars Technica article “Bigger is better: Quantum volume expresses computer’s limit” provides some historical context, as well as example QV calculation results.

## What is Quantum Volume

As noted in the Forbes article “Quantum Volume: A Yardstick To Measure The Performance Of Quantum Computers”, quantum volume measurement relates incremental advancements in quantum computer hardware on the roadmap to fault-tolerant quantum computing (FTQC) and quantum advantage. As noted in our article “Government Quantum Computing Initiatives: An In-Depth Exploration,” hardware providers can therefore flaunt QV scores as an indicator of technological leadership. Commercially, QV scores provide an easy way for corporate users to compare the available quantum computers.

## Understanding the Components of Quantum Volume

Despite the name and the use of square quantum circuits, Quantum Volume isn’t calculated like the volume of a cube. Instead, multiple statistical tests are performed. The features of a quantum computer that influence the outcomes of these tests include:

- The number of physical qubits available, with higher counts being better
- Gate and measurement error rates, with lower rates being better
- Qubit connectivity and the cross-talk between neighboring qubits

Connectivity indicates which qubits can be entangled directly. The quantum volume of electron based modalities, such as superconducting qubits, is limited because each physical qubit is directly connected to a limited number of neighboring qubits. QV therefore favors modalities such as neutral atoms, which have all-to-all connectivity.

*The Quantum Volume Formula*

The definition of Quantum Volume has evolved over the past few years. Circuit depth ‘d’ depends on the average error rate ‘ε’ of a two-qubit gate:

d=1/ϵ

However, if qubit connectivity is limited, SWAP gates may be needed to help implement some two-qubit gates. With ‘ε_phys’ as the average error rate of a two-qubit gate and ‘n’ as the number of qubits, the average error rate becomes:

ϵ=ϵ_"phys" +n_"SWAP" ⋅ϵ_"SWAP"

When ‘d < n,’ which means that the number of qubits exceeds the maximum circuit depth, adding qubits lowers the QV score:

V_Q=(max)┬(n≤N) min(n,d)^2

However, adding qubits exponentially increases the complexity of simulating circuits on classical computers. Therefore, QV was re-defined as an exponential of circuit size:

V_Q=2^QV

Using this definition with a square quantum circuit (width = depth), QV indicates the deepest circuit that can be implemented on the target quantum computer with a success likelihood exceeding 2/3 and a confidence interval surpassing 97.725%.

## Quantum Volume and Quantum Supremacy

“Quantum supremacy” is a demonstration that a quantum computer can solve a problem that no classical computer can solve in a reasonable timeframe. While these demonstrations are important, one shortcoming of these demonstrations is that they do not require quantum error correction (QEC). The accuracy of the results is not significant.

To solve real-world problems, however, accuracy matters. “Quantum advantage” would be the demonstration of a quantum computer solving a commercially-relevant problem faster than a classical computer with an accuracy that makes the results useful. Because quantum advantage is not yet possible, increases in Quantum Volume scores indicate that progress is being made toward achieving this goal. There is no set QV score that indicates quantum advantage is possible, but increases nonetheless indicate progress.