Average infidelity

One way of defining performance of a quantum process is through an infidelity measure. We use the average operational infidelity, as it is easy to calculate, a function of the operators only and easy to relate to other infidelity measures. We define it as follows:

Given an ideal unitary gate $U$, with corresponding map

\[\begin{align} \rho \mapsto \mathcal{U}(\rho) = U\rho U^\dagger \end{align}\]

and some realized map \(\mathcal{V}(\rho)\):

\[\begin{align} \rho \mapsto \mathcal{V}(\rho) \end{align}\]

We can define the error map as:

\[\begin{align} \rho \mapsto \mathcal{E}(\rho) = \mathcal{U}^{-1} \circ \mathcal{V}(\rho) \end{align}\]

The operational infidelity of the error map is based on the the norm of the Hilbert-Schmidt inner product, but the precise calculation will depend on the nature of the error map. In all cases it is defined in terms of the average operational fidelity:

\[\mathcal{I}_{av} = 1 - \mathcal{F}_{av}.\]

If the error map is deterministic and unitary, namely it can be represented as:

\[\begin{align} \mathcal{E}(\rho) = E \rho E^\dagger = U^\dagger V \rho V^\dagger U, \end{align}\]

where $V$ is the realized unitary, and $E$ is the error unitary then the operational fidelity is:

\[\mathcal{F}_{av} = \left| \frac{1}{D} \mathrm{Tr}[ E ] \right|^2.\]

If the error map is a completely positive and trace preserving, then it can be represented as a sum of Kraus operators:

\[\begin{align} \mathcal{E}(\rho) = \sum_i E_i \rho E_i^\dagger = \sum_i U^\dagger V_i \rho V_i^\dagger U, \end{align}\]

where $E_i$ are the Kraus operators for the error process, $V_i$ are the Kraus operators for the realized map then the average operational fidelity is:

\[\mathcal{F}_{av} = \sum_i \left| \frac{1}{D} \mathrm{Tr}[ E_i ] \right|^2.\]

If the error map is a stochastic unitary process:

\[\begin{align} \mathcal{E}(\rho) = \left\langle \tilde{E} \rho \tilde{E}^\dagger \right\rangle = \left \langle U^\dagger \tilde{V} \rho \tilde{V}^\dagger U \right\rangle, \end{align}\]

where $\tilde{E}$ is a random unitary matrix for the error process, $\tilde{V}$ is the random unitary matrix for the realized map, and here $\left\langle \cdot \right\rangle$ is the average with regard to the random unitary maps. Then the average operational fidelity is:

\[\mathcal{F}_{av} = \left\langle \left| \frac{1}{D} \mathrm{Tr}[ \tilde{E} ] \right|^2 \right\rangle.\]

Note these methods of calculating the average operational fidelity all reduce to one another in the appropriate limit.

Robust control


Here we define robust control. Let $H_{tot}(t)$ denote the total Hamiltonian for a system including all dynamical contributions from both control and noise interactions. In the limit of perfect control (no noise) this reduces to the control Hamiltonian $H_{tot}(t)\rightarrow H_{c}(t)$. Noisy and perfect system evolution is therefore determined by solving the Schrödinger equation

\[\begin{align} i\dot{U}_{tot}(t) &= H_{tot}(t)U_{tot}(t) &&\text{(noisy evolution)}\\ i\dot{U}_c(t) &= H_{c}(t)U_{c}(t) &&\text{(perfect evolution)}. \end{align}\]

Evolution over duration $\tau$ under $H_{tot}(t)$ generates the noisy operation $U_{tot}(\tau)$, while evolution under $H_{c}(t)$ generates the perfect operation $U_c(\tau)$, defining the control solution.

The control solution is said to be robust if, under $H_{c}(t)$, the operator-distance between $U_c(\tau)$ and $U_{tot}(\tau)$ is reduced in the ensemble-average over noise realizations.


Here we define a measure to quantify robustness of a given control solution against noise in the system. From the definition above this may be conditioned on an ensemble-average operator-distance between $U_c(\tau)$ and $U_{tot}(\tau)$. For this we use a modified Frobenius inner product to establish the following measure

\[\begin{align} \mathcal{F}_{robust}(\tau) = \left\langle\left|\frac{1}{\text{Tr}\left(P\right)}\text{Tr}\left(P U_{c}^\dagger (\tau) U_{tot}(\tau)P\right)\right|^2\right\rangle \end{align}\]

where the projection matrix $P$ has been incorporated to enable computation on a target subspace of the full Hilbert space, and the angle brackets $\langle\cdot\rangle$ denote an ensemble average over realizations of the noise process.

This measure uses the trace to express the operator distance between $U_{c}^\dagger$ and $U_{tot}$ in terms of how close $U_{c}^\dagger U_{tot}$ is to the identity $\mathbb{I}$. The projection matrix $P$ limits this measurement to the associated subspace via the transformation

\[\begin{align} U_{c}^\dagger U_{tot} \rightarrow P(U_{c}^\dagger U_{tot} )P. \end{align}\]

This definition for robustness is target-independent. That is, it only measures the degree to which the final operation, $U_{tot}$, deviates from the perfect operation, $U_{c}$, under noise interactions. It does not require the control solution to implement a particular target operation $U_{c}$. An additional measure constraining the particular final operation must be used to enforce this criterion.

Note, division by $\text{Tr}\left(P\right)$ in the above definition ensures that we recover unit fidelity in the event of perfect control (no noise). Namely, if $ U_{tot} = U_{c}$ we obtain

\[\begin{align} \mathcal{F}_{robust}(\tau) = \left\langle\left|\frac{1}{\text{Tr}\left(P\right)}\text{Tr}\left(P U_{c}^\dagger U_{c} P\right)\right|^2\right\rangle = \left\langle\left|\frac{1}{\text{Tr}\left(P\right)}\text{Tr}\left(P\mathbb{I} P\right)\right|^2\right\rangle =\left\langle\left|\frac{1}{\text{Tr}\left(P\right)}\text{Tr}\left(P\right)\right|^2\right\rangle =1. \end{align}\]

Note, for a $D$-dimensional Hilbert space, when $P=\mathbb{I}$ we obtain $\text{Tr}\left(P\right)=D$. In this case the metric reduces to

\[\begin{align} \mathcal{F}_{robust}(\tau) = \left\langle \left|\frac{1}{D}\text{Tr}\left(U_{c}^\dagger U_{tot}\right)\right|^2\right\rangle \end{align}\]

and evaluates robustness over the full Hilbert space.

Filter function infidelity

The measure for robustness defined above may be approximated using filter function framework. Specifically

\[\begin{align} \mathcal{I}_{robust} &= 1 - \mathcal{F}_{robust} \approx \frac{1}{2} \left(1 - \exp\left(-2 \sum_{k=1}^{p} \mathcal{O}_k\right)\right)\\ \mathcal{O}_{k} &= \frac{1}{2\pi} \int_{-\infty}^{\infty} S_{k}(\omega) F_{k}(\omega) d\omega \end{align}\]

where \(\mathcal{O}_{k}\) is the contribution from the $k$th noise channel, expressed as an overlap integral between the noise power spectral density, $S_{k}(\omega)$, and the associated filter function $F_{k}(\omega)$.

The filter function framework assumes a total Hamiltonian of the form

\[\begin{align} H_{tot}(t)=H_{c}(t)+ H_{n}(t) \end{align}\]

where the control Hamiltonian, $H_{c}(t)$, generates perfect evolution and the noise Hamiltonian, $H_{n}(t)$, generates undesirable interactions with relevant noise processes. These evolution paths are obtained by solving the Schrödinger equation

\[\begin{align} i\dot{U}_c(t) &= H_{c}(t)U_{c}(t) &&\text{(target evolution)},\\ i\dot{U}_{tot}(t) &= H_{tot}(t)U_{tot}(t) &&\text{(noisy evolution)}, \end{align}\]

The error process is then treated as a stochastic unitary map defined by the operator $\tilde{U} = U_{c}^\dagger U_{tot}$.

Noise-free infidelity

For some controls, the control itself may not realize the ideal target unitary. We refer to this as a noise-free infidelity, where we use the average operational infidelity for a unitary error process.

\[\begin{align} \mathcal{I}_{av}(\tau) = 1-\frac{1}{D^2} \left|\text{Tr}[U_{target}^\dagger(\tau) U_c(\tau)]\right|^2. \end{align}\]

This quantity measures the deviation between the target operation $U_{target}$, and a perfectly implemented control operation $U_c(\boldsymbol{v})$ (i.e. assuming no noise interactions). In general, in the absence of noise this value approaches zero, but it can become appreciable in the case where a complex optimized control solution is employed. Again, however, this value is typically many orders of magnitude smaller than the noise-induced infidelity calculated in the filter function framework.

Loss infidelity

If a qubit is coupled to an environment, such that it is undergoing a loss process described by a master equation of the form:

\[\frac{d\rho}{dt} = \frac{\gamma}{2} \mathcal{D}[\sigma](\rho) = \frac{\gamma}{2} \left( \sigma \rho \sigma^\dagger - \frac{1}{2}( \sigma^\dagger \sigma \rho + \rho \sigma^\dagger \sigma) \right)\]

If a process goes for a time $\tau$ then the average operational infidelity is

\[\mathcal{I}_{av,L} = 1 - \frac{1}{4} ( 1 + e^{- \gamma \tau/ 2})^2.\]

Total infidelity

The total infidelity is the exponential sum of the average operational infidelities. This is simply meant to be an indicative summary of fidelities, rather than a formal bound on the total infidelity of the process. Given a set of infidelities they are added then the smaller of the sum or 1 is taken:

\[\begin{align} \mathcal{I}_{av, T} = \text{min}(\sum_{k=1}^{p} \mathcal{I}_k, 1.0) \end{align}\]

Gate infidelity

Another common infidelity is the average gate infidelity, this can be related simply to the average operational infidelity using:

\[\mathcal{I}_{av, G} = \frac{D}{D+1} \mathcal{I}_{av}\]

This relation is derived from equation (12) in Quantum gate fidelity in terms of Choi matrices.

State infidelity

To quantify the performance of a Mølmer-Sørensen control operation, we use the state infidelity. This is given in terms of the overlap between the final two-qubit density matrix $\rho_{Q}$, produced by applying the generated control operation to the initial state $\vert 00 \rangle$, and one of the two possible maximally entangled target states

\[\begin{equation} \vert \Phi^{\pm} \rangle = \frac{1}{\sqrt{2}} ( \vert {00} \rangle \pm i \vert {11} \rangle ), \end{equation}\]

which are obtained from the application of the ideal Mølmer-Sørensen gate. In this case, the infidelity is explicitly given by:

\[\begin{align} \mathcal{I} &= 1 - \langle \Phi^{\pm} \vert \rho_{Q} \vert \Phi^{\pm} \rangle \\ &= 1 - \frac{1}{2} (P_0 + P_2 \pm 2\textrm{Im} [ \langle 00 \vert \rho_{Q} \vert 11 \rangle] ), \end{align}\]

with $P_0, \, P_2$ representing the two-qubit populations in $\vert 00 \rangle$ and $\vert 11 \rangle$, respectively. The infidelity is always calculated with respect to the entangled state ($\Phi^{+}$ or $ \Phi^{-}$) that matches the sign of the accumulated entangling phase produced by the control operation.

Worst-case error bound

The worst case error bound is derived from the average operational infidelity, through the average gate infidelity, and provides a bound on the operation error.

The average infidelity is an accessible quantity, both in terms of calculation and experimental measurements. However the average infidelity does not bound the true fidelity of an operation, and it can not be used directly to bound the total fidelity of a sequence of operations.

Fortunately, the average gate infidelity can be used to bound the the true error for an operation:

\[\mathscr{E}_{op} \le \mathscr{E}_{bound} = D \sqrt{\left(1 + \frac{1}{D} \right) \mathcal{I}_{av, G}}\]

Where the \(\mathcal{I}_{av}\) is the average infidelity and $E_{bound}$ is the worst case error bound.

The error of an operation $E_{op}$ is particularly useful when examining a sequence of operations. If a set of control solutions are used in a quantum computation the total error of the operation is bounded by the sum of the errors for each individual operation. Furthermore the probability that a quantum computation fails can be bounded by this total bound.

For more details on this relationship see Bounding quantum gate error rate based on reported average fidelity by Yuval R Sanders et al. 2015.

Total error

The worst-case error bound can be summed to give the total error for the full error process:

\[\mathscr{E}_{bound,T} = \sum_k \mathscr{E}_{bound,k}\]