Robust characterization of leakage errors

An important error mechanism in many experimental implementations of quantum information is leakage, that is, transitions into and out of the Hilbert space under consideration (e.g., an electron excitation to another energy level). Subsequent transitions back into the Hilbert space introduce a memory effect, making leakage a fundamentally non-Markovian process. Such leakage errors can be a substantial obstacle to fault-tolerant computation [1–3].

There are platform-dependent methods for characterizing leakage in many of the leading experimental approaches to quantum computation, such as ion trap qubits [4], superconducting qubits [5, 6] and quantum dots [7]. However, these approaches all have disadvantages such as being platform-dependent, scaling exponentially in the number of qubits, being sensitive to state-preparation and measurement errors (SPAM) or assuming a specific error model.

Randomized benchmarking (RB) [8–10] has been specifically developed to avoid all of these pitfalls at the cost of obtaining only partial information—namely, the average gate fidelity—about the errors in the absence of leakage. In the presence of leakage, the standard fidelity decay curve in RB breaks down [11], although the RB protocol can be modified to account for leakage errors [12].

We present a protocol that provides an estimate of the average leakage rate for coherent leakage over a given set of quantum gates. We consider computational and leakage spaces of arbitrary dimensions, so that our protocol can be applied to both physical and logical qudit systems. We demonstrate that our protocol produces reliable estimates of leakage rates through numerical simulations of our protocol for specific, adversarial, error models.

Note that after the protocol below first appeared online, an alternative heuristic protocol was presented in [13]. While the heuristic protocol applies to specific experimental scenarios, the current protocol is both fully rigorous and expressed in terms of platform-independent experimental capabilities.

Many experimental implementations of logical d₁-level qudits (typically ${d}_{1}=2$, giving a qubit) are embedded in a d-level quantum system by taking only the first d₁ levels $| 1\rangle ,\ldots ,| {d}_{1}\rangle$. Formally, we can consider a decomposition of the d-dimensional Hilbert space ${ \mathcal H }$ into two orthogonal subspaces ${{ \mathcal H }}_{1}={\rm{span}}(| 1\rangle ,\ldots ,| {d}_{1}\rangle )$ and ${{ \mathcal H }}_{2}={\rm{span}}(| {d}_{1}+1\rangle ,\ldots ,| d\rangle )$ of dimensions d₁ and ${d}_{2}=d-{d}_{1}$ respectively. Many stochastic and unitary processes on the physical space, that is, ${ \mathcal H }$, result in leakage from ${{ \mathcal H }}_{1}$ into ${{ \mathcal H }}_{2}$ and vice versa.

Within the broad framework of time-dependent trace-preserving Markovian noise on an extended Hilbert space ${ \mathcal H }$, any experimental implementation of a unitary G can be written as ${{ \mathcal E }}_{G}=G\;\circ \;{ \mathcal E }$ for some completely positive and trace-preserving map ${ \mathcal E }\;:{ \mathcal D }({ \mathcal H })\to { \mathcal D }({ \mathcal H })$, where ${ \mathcal D }({ \mathcal H })$ is the set of density operators (that is, self-adjoint operators with unit trace) acting on ${ \mathcal H }$ and ${ \mathcal A }\;\circ \;{ \mathcal B }$ denotes channel composition from left to right, that is, apply ${ \mathcal B }$ then ${ \mathcal A }$.

The net probability of a system described by a state $\rho \in { \mathcal D }({{ \mathcal H }}_{1})$ leaking out of the subspace ${{ \mathcal H }}_{1}$ and into an orthogonal subspace ${{ \mathcal H }}_{2}$ under a map ${ \mathcal E }$ is

$$\begin{eqnarray}&&{L}^{1\to 2}(\rho | { \mathcal E },{{ \mathcal H }}_{1})=\mathrm{Tr}[{{\bf{P}}}_{{{ \mathcal H }}_{2}}{ \mathcal E }(\rho )],\end{eqnarray} \tag{ 1 }$$

where ${{\bf{P}}}_{{{ \mathcal H }}_{{\rm{k}}}}$ is the projector onto ${{ \mathcal H }}_{{\rm{k}}}$, that is, ${{\bf{P}}}_{{{ \mathcal H }}_{{\rm{k}}}}| j\rangle =| j\rangle$ if $| j\rangle \in {{ \mathcal H }}_{{\rm{k}}}$ and ${{\bf{P}}}_{{{ \mathcal H }}_{{\rm{k}}}}| j\rangle =0$ if $| j\rangle \notin {{ \mathcal H }}_{{\rm{k}}}$. Similarly it is convenient to define the survival probability for the subspace ${{ \mathcal H }}_{1}$ as follows

$$\begin{eqnarray*}{S}^{1\to 1}(\rho | { \mathcal E },{{ \mathcal H }}_{1}) & = & \mathrm{Tr}[{{\bf{P}}}_{{{ \mathcal H }}_{1}}{ \mathcal E }(\rho )]\\ & = & 1-{L}^{1\to 2}(\rho | { \mathcal E },{{ \mathcal H }}_{1}),\end{eqnarray*}$$

where in the second line we have used ${{\bf{P}}}_{{{ \mathcal H }}_{1}}+{{\bf{P}}}_{{{ \mathcal H }}_{2}}={{\bf{P}}}_{{ \mathcal H }}$. These probabilities depend on the input state, but by the linearity of the trace and of quantum channels, the leakage rate averaged over all pure states in ${{ \mathcal H }}_{1}$ is simply the leakage probability for $\tfrac{1}{{d}_{1}}{{\bf{P}}}_{{{ \mathcal H }}_{1}}$, i.e., the maximally mixed state over ${{ \mathcal H }}_{1}$, and hence

$$\begin{eqnarray}{L}_{{\rm{avg}}}^{1\to 2}({ \mathcal E },{{ \mathcal H }}_{1}) & = & {L}^{1\to 2}\left(\tfrac{1}{{d}_{1}}{{\bf{P}}}_{{{ \mathcal H }}_{1}}| { \mathcal E },{{ \mathcal H }}_{1}\right)\\ & = & 1-\mathrm{Tr}\left[{{\bf{P}}}_{{{ \mathcal H }}_{1}}{ \mathcal E }\left(\tfrac{1}{{d}_{1}}{{\bf{P}}}_{{{ \mathcal H }}_{1}}\right)\right].\end{eqnarray} \tag{ 2 }$$

What is of particular interest here is the possibility of the quantum state re-entering ${{ \mathcal H }}_{1}$ at a later time: that is, leakage out of ${{ \mathcal H }}_{2}$ back into ${{ \mathcal H }}_{1}$. The coherent leakage back into ${{ \mathcal H }}_{1}$ is arguably more problematic for characterizing the quantum process in general and for fault-tolerance in particular because it introduces memory effects. Specifically, it makes the noise non-Markovian on the subspace ${{ \mathcal H }}_{1}$. To characterize the total impact of leakage both out of and back into the subspace ${{ \mathcal H }}_{1}$, we consider the sum of these two effects, averaged over input states in each subspace as follows

$$\begin{eqnarray}{L}_{{\rm{sum}}}({ \mathcal E },{{ \mathcal H }}_{1}) & = & {L}_{{\rm{avg}}}^{1\to 2}({ \mathcal E },{{ \mathcal H }}_{1})+{L}_{{\rm{avg}}}^{2\to 1}({ \mathcal E },{{ \mathcal H }}_{2})\\ & = & 1-{S}_{{\rm{sum}}}({ \mathcal E },{{ \mathcal H }}_{1}),\end{eqnarray} \tag{ 3 }$$

where the second line implicitly defines the associated survival rate accounting for both effects.

For trace-decreasing noise, there are multiple inequivalent definitions of leakage rates, depending on how the loss rate (i.e., decrease in trace) is incorporated. However, our protocol still allows the subspace survival rate ${S}_{{\rm{sum}}}({ \mathcal E },{{ \mathcal H }}_{1})$ defined in equation (3) to be estimated and the loss rate can be estimated via the protocol of [14].

We now present a protocol for characterizing the average subspace survival rate averaged over the noise ${ \mathcal E }$ in the experimental implementation of a set of operations ${\mathbb{G}}=\{U\oplus \mu V\;:U\in {\mathbb{U}},V\in {\mathbb{V}},\mu =\pm \}$, where ${\mathbb{U}}$ and ${\mathbb{V}}$ are unitary one-designs [15] on ${{ \mathcal H }}_{1}$ and ${{ \mathcal H }}_{2}$ respectively and ⊕ denotes the matrix direct sum, so that

$$\begin{eqnarray}(U\oplus \mu V)| j\rangle =\left\{\begin{array}{ll}U| j\rangle & {\rm{if}}\;{\rm{}}\quad | j\rangle \in {{ \mathcal H }}_{1}\\ \mu V| j\rangle & {\rm{if}}\;{\rm{}}\quad | j\rangle \in {{ \mathcal H }}_{2}.\end{array}\right.\end{eqnarray} \tag{ 4 }$$

Note that standard RB requires a unitary two-design, which is a strictly stronger requirement than a one-design. Note also that the phase factor μ is automatically included if the one-design is chosen to be a group with phase factors included. Moreover, these phase factors can be applied in whichever subspace is most convenient. For simplicity, we assume that the noise ${ \mathcal E }$ is time- and gate-independent. Perturbative results for time- and gate-dependent noise can be obtained by applying the approach of [16].

Our protocol for estimating ${L}_{{\rm{sum}}}({ \mathcal E },{{ \mathcal H }}_{1})$ is as follows.

• (1)  
  Choose a sequence length $m\in {\mathbb{N}}$.
• (2)  
  Choose a random sequence $\vec{G}=({G}_{1},\ldots ,{G}_{m})$ of m gates uniformly at random from ${\mathbb{G}}$.
• (3)  
  Prepare a state $\rho \approx {{ \mathcal P }}_{{{ \mathcal H }}_{1}}/{d}_{1}$.
• (4)  
  Apply the sequence of gates ${G}_{m}\ldots {G}_{1}$.
• (5)  
  Measure $Q\approx {{ \mathcal P }}_{{{ \mathcal H }}_{1}}$.
• (6)  
  Repeat steps 3–5 to estimate

  $$\begin{eqnarray}&&{Q}_{\vec{G}}=\mathrm{Tr}[Q{{ \mathcal G }}_{m}\;\circ \;{ \mathcal E }\;\circ \ldots \circ \;{{ \mathcal G }}_{1}\;\circ \;{ \mathcal E }(\rho )]\end{eqnarray} \tag{ 5 }$$

  to a desired precision where ${ \mathcal G }(\rho )=G\rho {G}^{\dagger }$.
• (7)  
  Repeat steps 2–6 to estimate

  $$\begin{eqnarray}&&{{\mathbb{E}}}_{\vec{G}}{Q}_{\vec{G}}=| {\mathbb{G}}{| }^{-m}\displaystyle \sum _{\vec{G}\in {{\mathbb{G}}}^{m}}{Q}_{\vec{G}}\end{eqnarray} \tag{ 6 }$$

  to a desired precision (see, e.g., [17] for methods to bound the number of sequences required to obtain a given precision).
• (8)  
  Repeat steps 1–7 for multiple m and fit to the decay curve in equation (7), derived below, to estimate ${L}_{{\rm{sum}}}({ \mathcal E },{{ \mathcal H }}_{1})$.

Note that this protocol differs from the loss protocol in [14] in that the operations require experimental control relative to the leakage levels to prevent leakage processes from accumulating coherently. This protocol differs from the RB protocol of [10] in that no inversion gate is applied immediately prior to the measurement so that the average state at the end of the circuit is a highly mixed state, which ensures that the estimated leakage rate is that of the maximally mixed state.

For noise that is trace-preserving on ${ \mathcal H }$, averaging the results over a number of random sequences with fixed m will give an estimate of

$$\begin{eqnarray}&&{{\mathbb{E}}}_{\vec{G}}{Q}_{\vec{G}}=A{[{S}_{{\rm{sum}}}({ \mathcal E },{{ \mathcal H }}_{1})]}^{m-1}+B,\end{eqnarray} \tag{ 7 }$$

where the constants A and B relate to SPAM. For noise that is not trace-preserving, an estimate of ${S}_{{\rm{sum}}}({ \mathcal E },{{ \mathcal H }}_{1})$ can still be obtained by fitting to the matrix exponential in equation (12).

We now derive the expression for ${{\mathbb{E}}}_{\vec{G}}{Q}_{\vec{G}}$, allowing for trace-decreasing noise unless otherwise specified. Uniformly averaging over all sequences $\vec{G}$ is equivalent to independently and uniformly averaging over the individual gates G_j, which gives

$$\begin{eqnarray}&&{{\mathbb{E}}}_{\vec{G}}{Q}_{\vec{G}}=\mathrm{Tr}[Q\bar{{ \mathcal G }}\;\circ \;{ \mathcal E }\;\circ \ldots \circ \;\bar{{ \mathcal G }}\;\circ \;{ \mathcal E }(\rho )],\end{eqnarray} \tag{ 8 }$$

where $\bar{{ \mathcal G }}=| {\mathbb{G}}{| }^{-1}{\sum }_{G\in {\mathbb{G}}}{ \mathcal G }$.

Let $\tau \in { \mathcal B }({ \mathcal H })$ be an arbitrary state express it in block form as

$$\begin{eqnarray}\mathrm{and}\tau =\left(\begin{array}{cc}{\tau }_{11} & {\tau }_{12}\\ {\tau }_{21} & {\tau }_{22}\end{array}\right),\end{eqnarray} \tag{ 9 }$$

where ${\tau }_{{jj}}$ is supported on ${{ \mathcal H }}_{j}$ and the off-diagonal terms encode coherences between the two subspaces. The action of $\bar{{ \mathcal G }}$ on τ is

$$\begin{eqnarray}\bar{{ \mathcal G }}(\tau ) & = & | {\mathbb{G}}{| }^{-1}\displaystyle \sum _{U,V,\mu }\left(\begin{array}{cc}U & 0\\ 0 & \mu V\end{array}\right)\left(\begin{array}{cc}{\tau }_{11} & {\tau }_{12}\\ {\tau }_{21} & {\tau }_{22}\end{array}\right)\left(\begin{array}{cc}{U}^{\dagger } & 0\\ 0 & \mu {V}^{\dagger }\end{array}\right)\\ & = & \left(\begin{array}{cc}| {\mathbb{U}}{| }^{-1}{\displaystyle \sum }_{U}U{\tau }_{11}{U}^{\dagger } & 0\\ 0 & | {\mathbb{V}}{| }^{-1}{\displaystyle \sum }_{V}V{\tau }_{22}{V}^{\dagger }\end{array}\right)\\ & = & \left(\begin{array}{cc}\tfrac{1}{{d}_{1}}{{\mathbb{I}}}_{{d}_{1}}\mathrm{Tr}{\tau }_{11} & 0\\ 0 & \tfrac{1}{{d}_{2}}{{\mathbb{I}}}_{{d}_{2}}\mathrm{Tr}{\tau }_{22}\end{array}\right)\\ & = & \displaystyle \sum _{j=1,2}\tfrac{1}{{d}_{j}}{{ \mathcal P }}_{{{ \mathcal H }}_{j}}\mathrm{Tr}{{ \mathcal P }}_{{{ \mathcal H }}_{j}}\tau ,\end{eqnarray} \tag{ 10 }$$

where averaging over the phase factors eliminated any coherences between the two subspaces and the penultimate line follows from the fact that ${\mathbb{U}}$ and ${\mathbb{V}}$ are unitary one-designs [15, 18]. Now let ${\tau }^{(x)}={\sum }_{j}{\tau }_{j}^{(x)}\tfrac{1}{{d}_{j}}{{ \mathcal P }}_{{{ \mathcal H }}_{j}}$ denote the state after the xth application of $\bar{{ \mathcal G }}$ for $x\gt 1$. Then

$$\begin{eqnarray}{\tau }^{(x+1)} & = & \bar{{ \mathcal G }}[{ \mathcal E }({\tau }^{(x)})]\\ & = & \displaystyle \sum _{j=1,2}{\tau }_{j}^{(x)}\bar{{ \mathcal G }}\left[{ \mathcal E }\left(\tfrac{1}{{d}_{j}}{{ \mathcal P }}_{{{ \mathcal H }}_{j}}\right)\right]\\ & = & \displaystyle \sum _{j,k=1,2}{\tau }_{j}^{(x)}\tfrac{1}{{d}_{k}}{{ \mathcal P }}_{{{ \mathcal H }}_{k}}\mathrm{Tr}\left[{{ \mathcal P }}_{{{ \mathcal H }}_{K}}{ \mathcal E }\left(\tfrac{1}{{d}_{j}}{{ \mathcal P }}_{{{ \mathcal H }}_{j}}\right)\right]\\ & = & \displaystyle \sum _{j,k=1,2}{\tau }_{j}^{(x)}\tfrac{1}{{d}_{k}}{{ \mathcal P }}_{{{ \mathcal H }}_{k}}S\left(\tfrac{1}{{d}_{j}}{{ \mathcal P }}_{{{ \mathcal H }}_{j}}| { \mathcal E },{{ \mathcal H }}_{k}\right),\end{eqnarray} \tag{ 11 }$$

where we have used the fact that the trace and quantum channels are linear. Rephrasing as a matrix equation and iterating gives

$$\begin{eqnarray}&&\left(\begin{array}{c}{\tau }_{1}^{(x+1)}\\ {\tau }_{2}^{(x+1)}\end{array}\right)={\bf{S}}\;\left(\begin{array}{c}{\tau }_{1}^{(x)}\\ {\tau }_{2}^{(x)}\end{array}\right)={{\bf{S}}}^{x}\;\left(\begin{array}{c}{\tau }_{1}^{(1)}\\ {\tau }_{2}^{(1)}\end{array}\right),\end{eqnarray} \tag{ 12 }$$

where ${{\bf{S}}}_{{ab}}=S\left(\tfrac{1}{{d}_{b}}{{ \mathcal P }}_{{{ \mathcal H }}_{b}}| { \mathcal E },{{ \mathcal H }}_{a}\right)$. All the entries of ${\bf{S}}$ are probabilities and the rows add to 1 as ${ \mathcal E }$ is trace-preserving. The eigenvalues of ${\bf{S}}$ are 1 and ${{\bf{S}}}_{11}+{{\bf{S}}}_{22}-1={S}_{{\rm{sum}}}({ \mathcal E },{{ \mathcal H }}_{1})$, so ${\bf{S}}$ is either the identity or has two distinct eigenvalues. Solving the above matrix equation for $m=x+1$ and substituting the result into equation (8) gives

$$\begin{eqnarray}&&{{\mathbb{E}}}_{\vec{G}}{Q}_{\vec{G}}={{AS}}_{{\rm{sum}}}{({ \mathcal E },{{ \mathcal H }}_{1})}^{m-1}+B,\end{eqnarray} \tag{ 13 }$$

where the constants A and B depend on the overlap between the SPAM and the corresponding eigenvector of ${\bf{S}}$. More specifically, the eigenvector of ${\bf{S}}$ corresponding to the constant term is always the maximally mixed state (under the assumption that the noise is trace-preserving), so that

$$\begin{eqnarray}&&B=\displaystyle \frac{1}{d}\mathrm{Tr}Q.\end{eqnarray} \tag{ 14 }$$

The other eigenvector will depend on the specific details of the noise channel, but will satisfy the constraint $0\leqslant A+B\leqslant 1$ as ${{\mathbb{E}}}_{\vec{G}}{Q}_{\vec{G}}$ is an average of probabilities for all m, including m = 1. For small leakage rates and SPAM errors, A + B should be almost 1.

The only role of the trace-preserving constraint in the above derivation is to enforce the constraint that the rows of ${\bf{S}}$ add to 1. Without this constraint, ${\bf{S}}$ can be non-normal. However, ${\bf{S}}$ can still be put in Jordan canonical form, which gives the decay curve

$$\begin{eqnarray}&&{{\mathbb{E}}}_{\vec{G}}{Q}_{\vec{G}}=A{\lambda }_{+}^{m-1}+B{\lambda }_{-}^{m-1}\end{eqnarray} \tag{ 15 }$$

if ${\bf{S}}$ is diagonalizable, where

$$\begin{eqnarray}&&{\lambda }_{\pm }=\tfrac{1}{2}({{\bf{S}}}_{11}+{{\bf{S}}}_{22})\pm \tfrac{1}{2}\sqrt{{({{\bf{S}}}_{11}-{{\bf{S}}}_{22})}^{2}+4{{\bf{S}}}_{21}{{\bf{S}}}_{12}}\end{eqnarray} \tag{ 16 }$$

are the eigenvalues of ${\bf{S}}$ and

$$\begin{eqnarray}&&{{\mathbb{E}}}_{\vec{G}}{Q}_{\vec{G}}={{AS}}_{{\rm{sum}}}{({ \mathcal E },{{ \mathcal H }}_{1})}^{m-1}+{{BmS}}_{{\rm{sum}}}{({ \mathcal E },{{ \mathcal H }}_{1})}^{m-2}\end{eqnarray} \tag{ 17 }$$

otherwise. Standard model selection techniques can then be used to determine which decay curve best fits the data.

While the subspace survival rate ${S}_{{\rm{sum}}}({ \mathcal E },{{ \mathcal H }}_{1})$ obtained via our protocol gives an important figure of merit for characterizing leakage out of a subspace ${{ \mathcal H }}_{1}$, another important figure of merit is the worst-case leakage

$$\begin{eqnarray}&&{L}_{\mathrm{max}}({ \mathcal E },{{ \mathcal H }}_{1})=\underset{{\rho }_{1}\in { \mathcal D }({{ \mathcal H }}_{1})}{\mathrm{max}}{L}^{1\to 2}({\rho }_{1}| { \mathcal E },{{ \mathcal H }}_{1})+\underset{{\rho }_{2}\in { \mathcal D }({{ \mathcal H }}_{2})}{\mathrm{max}}{L}^{2\to 1}({\rho }_{2}| { \mathcal E },{{ \mathcal H }}_{2}).\end{eqnarray} \tag{ 18 }$$

However, as a consequence of the following Proposition, we have

$$\begin{eqnarray}&&{L}_{\mathrm{max}}({ \mathcal E },{{ \mathcal H }}_{1})\leqslant \mathrm{max}\{{d}_{1},{d}_{2}\}{L}_{{\rm{sum}}}({ \mathcal E },{{ \mathcal H }}_{1}).\end{eqnarray} \tag{ 19 }$$

Moreover, this bound is saturated and so cannot be improved without further knowledge about the noise. The following proof is a straight-forward modification of the corresponding proposition for the worst-case loss in [14].

Proposition 1. For any quantum channel ${ \mathcal E }$ and state $\rho \in { \mathcal D }({{ \mathcal H }}_{1})$ for a d-dimensional system

$$\begin{eqnarray*}&&{L}^{1\to 2}(\rho | { \mathcal E },{{ \mathcal H }}_{1})\leqslant {d}_{1}{L}^{1\to 2}\left(\tfrac{1}{{d}_{1}}{{\bf{P}}}_{{{ \mathcal H }}_{1}}| { \mathcal E },{{ \mathcal H }}_{1}\right),\end{eqnarray*}$$

where d₁ is the dimension of ${{ \mathcal H }}_{1}$. Moreover, for all d and subspaces ${{ \mathcal H }}_{1}$ there exist channels ${ \mathcal E }$ and states $\rho$ that saturate this bound.

Proof. Let ρ and ${ \mathcal E }$ be arbitrary states of and channels for a d-dimensional system.

Let ${\rho }^{\prime }=({{\bf{P}}}_{{{ \mathcal H }}_{1}}-\rho /\mathrm{Tr}\rho )/({d}_{1}-1)$, which is a valid quantum state since it is Hermitian and positive-semidefinite by construction and has unit trace. Since ${\rho }^{\prime }$ is a valid quantum state, the probability of detecting a system in the subspace ${{ \mathcal H }}_{2}$ after applying ${ \mathcal E }$ to a system prepared in the initial state ${\rho }^{\prime }$ is a true probability and thus

$$\begin{eqnarray}0 & \leqslant & {L}^{1\to 2}({\rho }^{\prime }| { \mathcal E },{{ \mathcal H }}_{1})\\ & \leqslant & \tfrac{1}{{d}_{1}-1}\left[{d}_{1}{L}^{1\to 2}\left(\tfrac{1}{{d}_{1}}{{\bf{P}}}_{{{ \mathcal H }}_{1}}| { \mathcal E },{{ \mathcal H }}_{1}\right)-{L}^{1\to 2}(\rho | { \mathcal E },{{ \mathcal H }}_{1})\right],\end{eqnarray} \tag{ 20 }$$

where we have used the fact that quantum channels and the trace are linear. Rearranging gives the desired bound.

This bound is saturated for any d and ${{ \mathcal H }}_{1}\subsetneq {{ \mathcal H }}_{d}$ by the channel ${{ \mathcal X }}_{\alpha }^{j\leftrightarrow k}$ corresponding to conjugation by

$$\begin{eqnarray}&&{X}_{\alpha }^{j\leftrightarrow k}={{\mathbb{I}}}_{d}+(\mathrm{cos}\alpha -1)(| j\rangle \langle j| +| k\rangle \langle k| )+{\rm{i}}\;\mathrm{sin}\;\alpha (| j\rangle \langle k| +| k\rangle \langle j| )\end{eqnarray} \tag{ 21 }$$

for any $\alpha \in {\mathbb{R}}$ and $| j\rangle \in {{ \mathcal H }}_{1}$, $| k\rangle \notin {{ \mathcal H }}_{1}$. In particular, a quick calculation shows that

$$\begin{eqnarray}&&{\mathrm{sin}}^{2}\alpha ={L}^{1\to 2}(| j\rangle \langle j| | {{ \mathcal X }}_{\alpha }^{j\leftrightarrow k},{{ \mathcal H }}_{1})={d}_{1}{L}^{1\to 2}\left(\tfrac{1}{{d}_{1}}{{\bf{P}}}_{{{ \mathcal H }}_{1}}| {{ \mathcal X }}_{\alpha }^{j\leftrightarrow k},{{ \mathcal H }}_{1}\right).\end{eqnarray} \tag{ 22 }$$

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We now investigate the performance of our protocol under a variety of noise models and illustrate that our protocol enables a robust estimation of the leakage rate under both gate-independent and gate-dependent noise. We consider a two-level (qubit) subspace ${{ \mathcal H }}_{1}={\rm{span}}(| 0\rangle ,| 1\rangle )$ of a three-level system ${ \mathcal H }={\rm{span}}(| 0\rangle ,| 1\rangle ,| 2\rangle )$, which is the most relevant form of leakage in many experimental platforms We choose ${\mathbb{U}}$ to be the standard set of qubit Pauli operators and ${\mathbb{V}}=1$.

To maximize the variation in leakage rate over states (and consequently over random sequences), we model leakage using the unitary ${X}_{\alpha }^{1\leftrightarrow 3}$ defined in equation (21). Coherent leakage can arise either naturally or be a residual effect of imperfectly storing a qubit in the leakage level, a technique used to protect certain states while performing an operation in various implementations, including ion traps [19, 20]. We do not consider one-way leakage, such as amplitude damping out of the two-level subspace and into the state $| 2\rangle$ because, under this type of Markovian noise model the current protocol performs equivalently to that of [14] and so is already analyzed there.

We first consider the gate-independent noise model

$$\begin{eqnarray}&&{ \mathcal E }={{ \mathcal U }}_{\phi }\;\circ \;\delta {\mu }_{\theta }\;\circ \;{{ \mathcal X }}_{\alpha }^{1\leftrightarrow 3}\;\circ \;{{ \mathcal A }}_{\eta }^{2\to 1},\end{eqnarray} \tag{ 23 }$$

where ${{ \mathcal A }}_{\eta }^{2\to 1}$ is an amplitude damping channel within the two-level subspace with Kraus operators

$$\begin{eqnarray}{A}_{1,\eta } & = & | 0\rangle \langle 0| +\sqrt{1-\eta }| 1\rangle \langle 1| +| 2\rangle \langle 2| \\ {A}_{2,\eta } & = & \sqrt{\eta }| 0\rangle \langle 1| \end{eqnarray} \tag{ 24 }$$

and ${{ \mathcal X }}_{\alpha }^{1\leftrightarrow },{{ \mathcal U }}_{\phi }$ and $\delta {\mu }_{\theta }$ correspond to conjugation by the unitary operators

$$\begin{eqnarray}&&{X}_{\alpha }^{1\leftrightarrow 3},\ [\mathrm{cos}\phi \;{{\mathbb{I}}}_{2}+{\rm{i}}\;\mathrm{sin}\;\phi \;{{UZU}}^{\dagger }]\oplus 1,\delta {\mu }_{\theta }={{\mathbb{I}}}_{2}\oplus {{\rm{e}}}^{{\rm{i}}\theta },\end{eqnarray} \tag{ 25 }$$

where $U\in U(2)$ is uniformly Haar-random to introduce unitary errors in the qubit subspace and errors in the relative phase respectively. We model SPAM errors by perturbing the diagonal entries of ρ and Q by random real numbers in the interval $[0,0.05]$, applying ${{ \mathcal U }}_{0.05}$ and re-normalizing so that $\mathrm{Tr}\rho =1$ and $\mathrm{Tr}Q=2$. The numerical leakage decay curves are shown in figure 1 with the relevant parameters discussed in the caption, demonstrating robust performance for the above noise model. In particular, the accuracy of the estimate obtained from our protocol is essentially independent of the size of the unitary errors (the orange squares and green circles). Furthermore, we observe that as the amount of amplitude damping (within the qubit subspace) increases, the statistical fluctuations of the individual data points become suppressed (purple diamonds) leading to a more precise estimate of the leakage rate. The average leakage rate increases slightly with the amount of amplitude damping as amplitude damping moves population to the $| 0\rangle$ level which then leaks under ${X}_{\alpha }^{1\leftrightarrow 3}$.

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To illustrate that our protocol is robust to gate-dependent errors, we simulate our protocol under the gate-dependent noise model

$$\begin{eqnarray}&&{{ \mathcal E }}_{P,\nu }=\delta {{ \mathcal P }}_{\phi }\;\circ \;\delta {\mu }_{\nu \theta }\;\circ \;{{ \mathcal X }}_{\alpha (P,\nu )}^{1\leftrightarrow 3},\end{eqnarray} \tag{ 26 }$$

where $\{P\oplus \mu \;:P=I,X,Y,Z,\mu =\pm \}$ are the ideal gates, $\delta {\mu }_{\nu \theta }$ introduces a relative phase error that depends on the target gate and

$$\begin{eqnarray}&&\delta {{ \mathcal P }}_{\phi }=[\mathrm{cos}\phi \;{{\mathbb{I}}}_{2}+{\rm{i}}\;\mathrm{sin}\;\phi \;P]\oplus 1\end{eqnarray} \tag{ 27 }$$

gives an over-rotation by ϕ in the Pauli operator P. The SPAM errors are simulated in the same manner as for the gate-independent noise. The results of numerical simulations are plotted in figure 2 and are well-described by the decay curve from equation (7) derived under the assumption of gate-independent noise. The primary difference from the gate-independent results is that the fluctuations in the individual data points is increased to the point that the individual data points are not consistent with the fitted model (derived for gate-independent noise) to within the standard error of the mean, even when the number of random sequences is increased.

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In this paper, we have presented a protocol for characterizing average subspace survival rates under coherent leakage to an orthogonal subspace. Experimentally implementing our protocol yields a decay curve which can be fitted to our analytical expressions to obtain the average probability of a leakage event occurring. We have also demonstrated that the decay can be observed and fitted in practice through numerical simulations of leakage for specific, adversarial, error models.

The leakage decay curve is derived assuming that the noise is independent of the gate, although the numerical results demonstrate that the same curve holds even under gate-dependent noise. However, when the noise is strongly gate dependent, the fluctuations in the decay are increased substantially. An open problem is to determine whether the strength of the fluctuations can be used to quantify the gate-dependence or the variation in leakage over states.

Our protocol is scalable and robust against SPAM. While only a unitary one-design is required for the current protocol, it can be applied in conjunction with standard RB to determine both the average leakage rate and the average gate infidelity over a unitary two-design such as the Clifford group.

The authors acknowledge helpful discussions with S Flammia, C Granade and T Monz. This research was supported by the US Army Research Office through grant W911NF-14-1-0103, CIFAR, the Government of Ontario, and the Government of Canada through NSERC and Industry Canada.