Fidelities and errors
FOM for classical probability distributions
Quantum states and processes cannot be observed directly, but measurements can be used as a proxy to get informations about these properties. Repeating the preparation and measurement of the same quantum state results as a probability associated to each state that can be observed. Quantum processes act on states and can then be measured to extract informations about these processes.
In both case, a probability distribution is obtained for each actual implementation \(\mathbf{p}\) and can be compared to an ideal target implementation \(\mathbf{\widetilde{p}}\) representing a noiseless quantum state or process (denoted with \(\sim\) character). Classical figure of merits comparing probability distributions can be used to evaluate how similar or different are two distributions. Among these FOMs, we outline those that are considered as metrics.
To be considered a metric (in the mathematical sense), a FOM \(D(\mathbf{p}, \mathbf{q})\) evaluated on ditributions \(\mathbf{p}\) and \(\mathbf{q}\) must:
- be symmetric: \(D(\textbf{p}, \textbf{q}) = D(\textbf{q}, \textbf{p})\).
- satisfy triangle inequality: \(D(\textbf{p}, \textbf{r}) \leq D(\textbf{p}, \textbf{q}) + D(\textbf{q}, \textbf{r})\).
- be null when equality: \(D(\textbf{p}, \textbf{q}) = 0\) if and only if \(\textbf{p}= \textbf{q}\). The reader may refer to [1] for an interesting discussion on the properties of metrics.
The next table summarizes the different FOMs that can be evaluated with classical distributions. It is assumed that the probability distribution is established over \(2^n\) different possible bitstrings noted \(\{0, 1\}^n\). The probability associated to each bitstring \(x\) is given by the probability distribution \(\mathbf{p}\) as \(\mathbf{p}(x)\).
| Name | Formula | Value if \(\mathbf{p} = \mathbf{\widetilde{p}}\) | Bounds | Is a metric ? | Operational interpretation |
|---|---|---|---|---|---|
| \(L_1\) distance aka. Manhattan distance |
$$L_1 \left( \mathbf{p}, \mathbf{\widetilde{p}} \right) = \left\lVert \mathbf{p} - \mathbf{\widetilde{p}} \right\rVert_1$$$$ = \sum_{x \in \{0,1\}^n} \left| \mathbf{p}(x) - \mathbf{\widetilde{p}}(x) \right|$$ | 0 | $$[0, 2]$$ | yes | |
| Total variation distance aka. Kolmogorov distance aka. Normalized \(L_1\) distance aka. Trace distance |
$$d_\mathrm{TV}\left(\mathbf{p}, \mathbf{\widetilde{p}}\right) = \frac{1}{2} L_1 \left( \mathbf{p}, \mathbf{\widetilde{p}} \right)$$ | 0 | $$[0, 1]$$ | yes | Single-shot discrimination error rate |
| Bhattacharya coefficient aka. Statistical overlap |
$$BC \left(\mathbf{p}, \mathbf{\widetilde{p}} \right) = \sum_{x \in \{0,1\}^n} \sqrt{\mathbf{p}(x) \mathbf{\widetilde{p}}(x)}$$ | 1 | $$[0, 1]$$ | N/M | Overlap measure between two distributions |
| Hellinger fidelity* aka. Classical fidelity |
$$F \left(\mathbf{p}, \mathbf{\widetilde{p}} \right) = \left( \sum_{x \in \{0,1\}^n} \sqrt{\mathbf{p}(x) \mathbf{\widetilde{p}}(x)} \right)^2$$ | 1 | $$[0, 1]$$ | no | |
| Hellinger distance [2] | $$d_\mathrm{H}\left(\mathbf{p}, \mathbf{\widetilde{p}} \right) = \sqrt{1-\sqrt{F\left(\mathbf{p}, \mathbf{\widetilde{p}}\right)}}$$ | 0 | $$[0, 1]$$ | yes | |
| Bhattacharya distance | $$d_\mathrm{B}\left(\mathbf{p}, \mathbf{\widetilde{p}} \right) = \frac{1}{2} \ln \left(BC \left(\mathbf{p}, \mathbf{\widetilde{p}} \right) \right)$$ | 0 | $$[0, \infty]$$ | no, do not satisfy triangle inequality | Metric measuring the overlap |
| Kullback-Leibler deviation [3] aka. Relative entropy |
$$d_\mathrm{KL}\left(\mathbf{p} || \mathbf{\widetilde{p}} \right) = \sum_{x \in \{0,1\}^n} \mathbf{p}(x) \log \left( \frac{\mathbf{p}(x)}{\mathbf{\widetilde{p}}(x)} \right)$$ | 0 | $$[0, \infty]$$ | no, do not satisfy triangle inequality | Many interpretations in statistics, inference … [4] |
| Entropy aka. Shannon entropy |
$$H\left(\mathbf{\widetilde{p}}\right) = -\sum_{x \in \{0,1\}^n} \mathbf{\widetilde{p}}(x) \log \left(\mathbf{\widetilde{p}}(x) \right)$$ | N/M | $$[0, \infty]$$ | N/M | |
| Cross-entropy | $$H(\mathbf{p}, \mathbf{\widetilde{p}}) = H\left(\mathbf{\widetilde{p}}\right) + d_{KL}\left(\mathbf{p} || \mathbf{\widetilde{p}}\right)$$ $$= -\sum_{x \in \{0,1\}^n} \mathbf{p}(x) \log \left(\mathbf{\widetilde{p}}(x) \right)$$ | Min val when \(\mathbf{p} = \mathbf{\widetilde{p}}\) | $$[0, \infty]$$ | no | |
| Linear Cross-entropy | $$H_\mathrm{lin}\left(\mathbf{p},\mathbf{\widetilde{p}}\right) = 2^n \left(\sum_{x \in \{0,1\}^n} \mathbf{p}(x) \mathbf{\widetilde{p}}(x)\right) – 1$$ | 1 | $$[0, 1]$$ | no |
*Sometimes the fidelity can be reported as the square root of the quantities reported here. We prefer this representation as it can be deirectly linked to the success probability of a quantum computation [1].
FOM for quantum states
The matrix density operator \(\rho=\sum_i p_i \ket{\psi_i} \bra{\psi_i}\) is used to model uncertainty about the knowledge of a quantum state. Each state \(\ket{\psi_i}\) is a pure state associated with the probability \(p_i\). We continue using the same notation where \(\widetilde{\rho}\) represents the ideal target quantum state and \(\rho\) the actual quantum state.
Assessing the distinguishability between quantum states strongly depends on the measures being done on the quantum states. For this reason, the measurement (Positive Operator-valued Measure) that maximizes the distinguishability between the two quantum states \(\rho\) and \(\widetilde{\rho}\) is always chosen. In this way, results obtained from these measurements are used to build metrics that assess an upper bound on the error rate of a measurement done on the quantum state. These metrics can be used to assess the performance of state preparations.
In the following table, the trace operator \(\Tr\) on a matrix \(A\) is defined as : \(\Tr(A) = \sum_{i} a_{ii}\).
| Name | Formula | Value if \(\mathbf{\rho = \widetilde{\rho}}\) | Bounds | Is a metric ? | Interpretation |
|---|---|---|---|---|---|
| Trace distance [5] | $$d_{tr} \left(\rho, \widetilde{\rho} \right) = \frac{1}{2} \left\lVert \rho - \widetilde{\rho} \right\rVert_1$$$$= \frac{1}{2} \Tr \left|\rho - \widetilde{\rho} \right|$$ | 0 | $$[0, 1]$$ | yes | inherits properties from classical total variation distance |
| State fidelity [5] (two pure states) |
$$\rho =\ket{\psi}\bra{\psi}, \widetilde{\rho} =\ket{\phi}\bra{\phi}$$$$F\left(\rho, \widetilde{\rho}\right) = |\braket{\psi | \phi}|^2$$ | 1 | $$[0, 1]$$ | no | Proxy to success probability of a quantum computation |
| State fidelity [5] (target is pure) (the other is mixed) |
$$\rho = \sum_i p_i \ket{\psi_i} \bra{\psi_i}, \widetilde{\rho} =\ket{\phi}\bra{\phi}$$$$F\left(\rho, \widetilde{\rho}\right) = \braket{\phi | \rho | \phi} = \Tr \left(\rho \ket{\phi}\bra{\phi} \right)$$ | 1 | $$[0, 1]$$ | no | Proxy to success probability of a quantum computation |
| State fidelity* [5] (with two mixed states) |
$$F \left(\rho, \widetilde{\rho} \right) = \left( \Tr \sqrt{\sqrt{\rho} \widetilde{\rho} \sqrt{\rho}} \right)^2$$ | 1 | $$[0, 1]$$ | no | Proxy to success probability of a quantum computation |
| Bures distance [1] (finite version of Bures metric) |
$$B \left( \rho, \widetilde{\rho} \right) = \sqrt{2-2\sqrt{F \left( \rho, \widetilde{\rho} \right)}}$$ | 0 | $$[0, \sqrt{2}]$$ | yes | Turn the fidelity to a euclidean distance between two pure states |
| Bures angle [1] aka. quantum angle |
$$A \left( \rho, \widetilde{\rho} \right) = \arccos \sqrt{F \left( \rho, \widetilde{\rho} \right)}$$ | 0 | $$[0, 2\pi]$$ | yes | Turn the fidelity to an angle between two pure states |
| State infidelity | $$\epsilon_{inf} = 1-F$$ | 0 | $$[0, 1]$$ | yes | Proxy to failure probability of a quantum computation |
| Quantum relative entropy [5] | $$S \left(\rho || \widetilde{\rho}\right) = \Tr \left( \rho \log(\rho) \right) - \Tr \left( \rho \log \left(\widetilde{\rho}\right) \right)$$ | 0 | $$[0, \infty]$$ | no, do not satisfy triangle inequality |
*Sometimes the fidelity can be reported as the square root of the quantities reported here. We prefer this representation as it can be deirectly linked to the success probability of a quantum computation [1].
FOM for quantum processes
Quantum processes (gates) define operations that transform the quantum state to another quantum state. These operations need to be accurate, and metrics have been defined to measure the quality of implementation of these processes. The evolution of the noisy quantum computer can be described with two main components:
- The quantum state: the system of interest used in the computation \(\rho\).
- The environment in which the quantum state evolves \(\rho_{env}\).
The evolution of the quantum state can be seen as a joint evolution of the state of interest and its environment (see. figure a). This evolution can be described by a single unitary \(U\) acting over both systems. At the end of the evolution, the quantum state \(\rho\) is measured using a projective measurement. This projective measurement also act over the environment as a joint measurement.
The map \(\rho \rightarrow M(\rho)\) represents a quantum operation if it is a Complete Positive Trace Preserving (CPTP) map. CP means that \(M(\rho)\) must be positive semi-definite (i.e., no event with a negative output probability distribution). TP means that each measurement outcome probability adds up to 1.
Benchmarking quantum processing involves comparing ideal implementations of the map \(\widetilde{M}\) to the actual implementation \(M\). In general, these maps are directly called quantum gates.
| Name | Formula | Value if \(\rho = \widetilde{\rho}\) | Bounds | Is a metric ? | Interpretation |
|---|---|---|---|---|---|
| Diamond distance [6] [7] | $$d_\mathrm{\diamond} \left(M, \widetilde{M} \right) = \frac{1}{2} \left\lVert M-\widetilde{M} \right\rVert_\mathrm{\diamond} $$$$ = max_\rho \left\lVert (M \otimes I)\rho - \left(\widetilde{M} \otimes I \right) \rho \right\rVert_1$$ | 0 | $$[0, 1]$$ | no | Achievable upper bound on the probability of distinguishing \(M\) from \(\widetilde{M}\) |
| Jamiołkowski trace distance [8] [1] | $$d_\mathrm{tr\_J}\left(M, \widetilde{M}\right) = \frac{1}{2} \left\lVert \left(M_A \otimes I_B - \widetilde{M}_A \otimes I_B \right) \ket{\psi_{AB}} \bra{\psi_{AB}} \right\lVert_1$$ | 0 | $$[0, 1]$$ | yes | Provide a lower bound on \(d_\mathrm{\diamond}\) |
| Average gate fidelity [5] | $$F_\mathrm{AVG} \left (M, \widetilde{M} \right) = \int F \left( M( \ket{\psi} \bra{\psi}), \widetilde{M}(\ket{\psi} \bra{\psi}) \right) d\psi$$ | 0 | $$[0, 1]$$ | no | Inherits quantum state fidelity properties Averaged over Haar measure Used in RB |
| Average gate infidelityaka. Average error rate* | $$\epsilon_{M} \left(M, \widetilde{M} \right) = 1 - F_\mathrm{AVG} \left(M, \widetilde{M} \right)$$ | 1 | $$[0, 1]$$ | no | error rate |
| Entanglement fidelity [9] | $$F_e \left(M, \widetilde{M} \right) = F \left( (M \otimes I) ( \ket{\psi} \bra{\psi}), \left( \widetilde{M} \otimes I \right)(\ket{\psi} \bra{\psi}) \right)$$ | 0 | $$[0, 1]$$ | no | Inherits quantum state fidelity properties |
| Entanglement infidelity | $$\epsilon_{e} \left(M, \widetilde{M} \right) = 1 – F_\mathrm{e} \left(M, \widetilde{M} \right)$$ | 1 | $$[0, 1]$$ | no | error rate |
| Stabilized minimum fidelity [1] | $$F_\mathrm{stab} \left( M, \widetilde{M} \right) = \min_{\psi_{AB}} F \left( \left( M_A \otimes I_B \right) (\ket{\psi} \bra{\psi}, \left( \widetilde{M_A} \otimes I_B \right)(\ket{\psi} \bra{\psi}) \right)$$ | 0 | $$[0, 1]$$ | no | Inherits quantum state fidelity properties |
*The average gate infidelity is often called the average gate error rate. However, in some cases, a distinction is done between these two quantities, as in quantum error correction for rigorously linking the error correction threshold to the average gate infidelity [10].
Extra references
For further reading on error measures of quantum states and quantum processing, the reader may refer to [1]. A nice recent tutorial discuss most of these metrics [11]. The reader can also refer to [5] for an introduction on distance measures.
References
- [1]A. Gilchrist, N. K. Langford, and M. A. Nielsen, “Distance measures to compare real and ideal quantum processes,” Physical Review A, vol. 71, no. 6, Jun. 2005, doi: 10.1103/physreva.71.062310. [Online]. Available at: http://dx.doi.org/10.1103/PhysRevA.71.062310
- [2]E. Hellinger, “Neue Begründung der Theorie quadratischer Formen von unendlichvielen Veränderlichen.,” Journal für die reine und angewandte Mathematik, vol. 1909, no. 136, pp. 210–271, Jul. 1909, doi: 10.1515/crll.1909.136.210. [Online]. Available at: http://dx.doi.org/10.1515/crll.1909.136.210
- [3]S. Kullback and R. A. Leibler, “On Information and Sufficiency,” The Annals of Mathematical Statistics, vol. 22, no. 1, pp. 79–86, Mar. 1951, doi: 10.1214/aoms/1177729694. [Online]. Available at: http://dx.doi.org/10.1214/aoms/1177729694
- [4]Wikipedia, “Kullback–Leibler divergence interpretations.” 2025 [Online]. Available at: https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence#Interpretations
- [5]M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information. Cambridge university press, 2010.
- [6]A. Y. Kitaev, “Quantum computations: algorithms and error correction,” Russian Mathematical Surveys, vol. 52, no. 6, pp. 1191–1249, Dec. 1997, doi: 10.1070/rm1997v052n06abeh002155. [Online]. Available at: http://dx.doi.org/10.1070/RM1997v052n06ABEH002155
- [7]D. Aharonov, A. Kitaev, and N. Nisan, “Quantum circuits with mixed states,” in Proceedings of the thirtieth annual ACM symposium on Theory of computing, 1998, pp. 20–30.
- [8]A. Jamiołkowski, “Linear transformations which preserve trace and positive semidefiniteness of operators,” Reports on Mathematical Physics, vol. 3, no. 4, pp. 275–278, Dec. 1972, doi: 10.1016/0034-4877(72)90011-0. [Online]. Available at: http://dx.doi.org/10.1016/0034-4877(72)90011-0
- [9]B. Schumacher, “Sending entanglement through noisy quantum channels,” Physical Review A, vol. 54, no. 4, p. 2614, 1996.
- [10]Y. R. Sanders, J. J. Wallman, and B. C. Sanders, “Bounding quantum gate error rate based on reported average fidelity,” New Journal of Physics, vol. 18, no. 1, p. 012002, Dec. 2015, doi: 10.1088/1367-2630/18/1/012002. [Online]. Available at: http://dx.doi.org/10.1088/1367-2630/18/1/012002
- [11]A. Hashim et al., “A Practical Introduction to Benchmarking and Characterization of Quantum Computers.” arXiv, 2024 [Online]. Available at: https://arxiv.org/abs/2408.12064