Hardware Metrics Table

A hardware metric has meaning only together with its definition, measurement procedure, operating context, and uncertainty. This table supplies a translation guide rather than a leaderboard. Metrics for Quantum Hardware explains how metrics enter a task-level assessment, while Device Characterization explains what experiments can identify.

Coherence, speed, and control quality

Quantity	Definition and comparison conditions
Population relaxation time T₁	Exponential population relaxation toward equilibrium; time units Temperature, reset, fitted equilibrium population, sequence, and nonexponential residuals.
Ramsey coherence time T₂*	Envelope decay in a specified free-precession experiment; time units Detuning drift, averaging window, envelope model, and removal of known oscillations.
Echo coherence time	Envelope decay after a specified refocusing sequence; time units Pulse sequence and spacing; echo and Ramsey values probe different noise filtering.
Average gate fidelity	$F_{\mathrm{avg}}(\mathcal E,U)$ defined below; dimensionless Target gate, Hilbert-space dimension, estimator, leakage treatment, and simultaneous-operation context.
Readout assignment fidelity	Mean probability of assigning the correct basis label, using the stated preparation priors Prepared-state errors, priors, discrimination threshold, integration time, and correlated errors.
Gate duration	Elapsed control interval; seconds Pulse tails, idles, transport, reset, and whether concurrent gates are allowed.
Error-correction cycle time	Duration of the declared syndrome round; seconds Which checks, measurement/reset times, scheduling, and decoder/feedforward latency.
Connectivity	Graph of directly available interactions, possibly weighted or time dependent Directionality, parallelism restrictions, native gate family, and movement costs.
Leakage probability	Population outside the declared computational subspace Preparation, number of gates or cycles, detection efficiency, and return processes.
Crosstalk	Change in a specified operation or observable caused by other active controls Spectator states, simultaneous schedule, measured response, and baseline; there is no universal scalar.
Logical failure probability	Probability of a defined logical error per operation, round, or full task Code, decoder, noise context, conditioning, duration, and failure event.

For the population row, $P_1(t)-P_1(\infty)\propto e^{-t/T_1}$ defines the exponential model. For uniform basis priors, the readout assignment fidelity is

$$F_{\mathrm{assign}}=\frac1d\sum_j P(\widehat j=j\mid j).$$

The relation $T_2^{-1}=(2T_1)^{-1}+T_\phi^{-1}$ applies to an exponential relaxation-plus-pure-dephasing model. In that model $T_2\leq2T_1$. Fitted Ramsey or echo constants from different filters and operating conditions should not be inserted into this relation without justification. See Dephasing and Amplitude Damping and Dynamical Decoupling.

Average fidelity and worst-case error

For a CPTP map on a fixed d-dimensional space and target unitary U, use the squared state-overlap convention:

$$\begin{aligned} F_{\mathrm{avg}}(\mathcal E,U) &=\int d\psi\,\langle\psi|U^\dagger\rho_{\mathrm{out}}(\psi)U|\psi\rangle,\\ \rho_{\mathrm{out}}(\psi)&=\mathcal E(\lvert\psi\rangle\langle\psi\rvert). \end{aligned}$$

The integral uses normalized Haar measure. If $F_e$ is entanglement fidelity of $U^\dagger\circ\mathcal E$ evaluated on the maximally mixed input, Nielsen’s relation is

$$F_{\mathrm{avg}}=\frac{dF_e+1}{d+1}.$$

A process fidelity agrees with $F_e$ only under compatible normalization and target conventions. Neither quantity is a worst-case error probability. Leakage, postselection, or input-dependent loss requires an enlarged-space or trace-decreasing treatment instead of automatically applying the CPTP fixed-space formula.

For the coherent qubit error $U_\theta=e^{-i\theta Z/2}$ relative to the identity,

$$1-F_{\mathrm{avg}}=\frac23\sin^2(\theta/2) \simeq\frac{\theta^2}{6}.$$

The half diamond distance from the identity channel is $|\sin(\theta/2)|$, linear in small $|\theta|$. Repeating this coherent error R times accumulates angle $R\theta$, so a small one-gate average infidelity need not mean a small error for a long sequence. Noise in Quantum Information develops that distinction.

Convert metrics only within a model

In the gate-independent depolarizing model for randomized benchmarking, a fitted decay parameter α gives average infidelity $r=(d-1)(1-\alpha)/d$ per sampled operation. A sampled Clifford can contain several native gates; dividing r by a mean gate count is an extra approximation. Gate dependence and representation choices further complicate interpreting a fitted RB rate as an average of individual gate infidelities, as Proctor and collaborators explain. See Randomized Benchmarking.

For independent task-ending failures of probability $p_L$ per round, surviving R rounds has probability $(1-p_L)^R$, giving failure $1-(1-p_L)^R\simeq Rp_L$ when $Rp_L\ll1$. This model does not describe every logical observable: two logical flips can cancel in a final parity measurement, and correlated events need a different model. Report the actual event and estimator before using a per-round number to predict a full algorithm.

Compare measurements under stated conditions

A reproducible metric should state the carrier or device subset, operating point and date, pulse or circuit family, simultaneous-operation context, shots and sequences, estimator, uncertainty, and accepted or discarded data. Report distributions across devices when available; the best element is not representative of a processor. Use Reporting Standards and Logical Benchmarking for worked comparisons.

References

• Nielsen, Michael A., “A simple formula for the average gate fidelity of a quantum dynamical operation,” Physics Letters A 303, 249–252 (2002), arXiv:quant-ph/0205035 — fidelity normalization and entanglement-fidelity relation.
• Proctor, Timothy, Kenneth Rudinger, Kevin Young, Mohan Sarovar, and Robin Blume-Kohout, “What randomized benchmarking actually measures,” Physical Review Letters 119, 130502 (2017), arXiv:1702.01853 — qualifications in interpreting randomized-benchmarking decay.
• Watrous, John, The Theory of Quantum Information, Cambridge University Press, 2018, author manuscript and errata — channel distance and state-overlap definitions.