# Quantum volume

Quantum volume is a metric that measures the capabilities and error rates of a quantum computer. It expresses the maximum size of square quantum circuits that can be implemented successfully by the computer. The form of the circuits is independent from the quantum computer architecture, but compiler can transform and optimize it to take advantage of the computer's features. Thus, quantum volumes for different architectures can be compared.

In 2020, the highest achieved quantum volume (per § IBM's modified definition) rose from 32 for IBM's computer "Raleigh" to 128 for Honeywell's "H1", i.e. quantum circuits of size up to 7×7 have been implemented successfully. More recently in 2021, Honeywell's "H1", first achieved a measured quantum volume of 512., and within six months doubled that performance to 1024.

## Introduction

Quantum computers are difficult to compare. Quantum volume is a single number designed to show all around performance. It is a measurement and not a calculation, and takes into account several features of a quantum computer, starting with its number of qubits—other measures used are gate and measurement errors, crosstalk and connectivity.

IBM introduced the Quantum Volume metric  because a classical computer’s transistor count and a quantum computer’s quantum bit count aren’t the same. Qubits decohere with a resulting loss of performance so a few fault tolerant bits are more valuable as a performance measure than a larger number of noisy, error-prone qubits.

Generally, the larger the quantum volume, the more complex the problems a quantum computer can solve.

## Definition

The quantum volume of a quantum computer is defined by Nikolaj Moll et al. It depends on the number of qubits N as well as the number of steps that can be executed, the circuit depth d

${\tilde {V}}_{Q}=\min[N,d(N)]^{2}.$ The circuit depth depends on the effective error rate $\epsilon _{\mathrm {eff} }$ as

$d\simeq {\frac {1}{N\epsilon _{\mathrm {eff} }}}.$ The effective error rate $\epsilon _{\mathrm {eff} }$ is defined as the average error rate of a two-qubit gate. If the physical two-qubit gates do not have all-to-all connectivity, additional SWAP gates may be needed to implement an arbitrary two-qubit gate and $\epsilon _{\mathrm {eff} }>\epsilon$ , where $\epsilon$ is the error rate of the physical two-qubit gates. If more complex hardware gates are available, such as the three-qubit Toffoli gate, it is possible that $\epsilon _{\mathrm {eff} }<\epsilon$ .

The allowable circuit depth decreases when more qubits with the same effective error rate are added. So with these definitions, as soon as $d(N) , the quantum volume goes down if more qubits are added. To run an algorithm that only requires $n qubits on an N-qubit machine, it could be beneficial to select a subset of qubits with good connectivity. For this case, Moll et al.  give a refined definition of quantum volume.

$V_{Q}=\max _{n where the maximum is taken over an arbitrary choice of n qubits.

### IBM's modified definition

The IBM's researchers modified the quantum volume definition to be an exponential of the circuit size, stating that it corresponds to the complexity of simulating the circuit on a classical computer:

$\log _{2}V_{Q}={\underset {n\leq N}{\operatorname {arg\,max} }}\left\{\min \left[n,d(n)\right]\right\}$ ### Algorithmic Qubits (AQ)

Algorithmic Qubits is the level equivalent of quantum volume, introduced by IonQ, and defined as $\log _{2}$ $V_{Q}$ .

IonQ announced a quantum computer with a claimed AQ of 22 (quantum volume of 4,194,304) in October 2020, although these numbers have not been empirically verified.

## Achievement history

Date Quantum volume[a]
(circuit size)
Manufacturer Notes
2020, January 32 (5×5) IBM "Raleigh" (28 qubits)
2020, June 64 (6×6) Honeywell 6 qubits
2020, August 64 (6×6) IBM Falcon r4 "Montreal" (27 qubits)
2020, November 128 (7×7) Honeywell "System Model H1" (10 qubits)
2020, December 128 (7×7) IBM Falcon r4 "Montreal" (27 qubits)
2021, March 512 (9×9) Honeywell "System Model H1" (10 qubits)
2021, July 1024 (10x10) Honeywell "Honeywell System H1" (10 qubits) 
2021, December 2048 (11x11) Quantinuum (previously Honeywell) "Quantinuum System Model H1-2" (12 qubits) 
2022, April 256 (8×8) IBM Falcon r10 "Prague" (27 qubits) 
2022, April 4096 (12x12) Quantinuum (previously Honeywell) "Quantinuum System Model H1-2" (12 qubits) 
2022, May 512 (9×9) IBM Falcon r10 "Prague" (27 qubits)