Quantum leakage detection using a model-independent dimension witness

Users of quantum computers must be able to confirm they are functioning as intended, even when they are remotely accessed. In particular, if the Hilbert space dimension of the quantum computer components is not as advertised—for instance, if they suffer leakage—errors can ensue and protocols may be rendered insecure. We refine the method of delayed vectors, adapted from classical chaos theory to quantum systems, and apply it remotely on the IBMQ platform—a quantum computer composed of transmon qubits. The method witnesses, in a model-independent fashion, dynamical signatures of higher-dimensional processes. We present evidence, under mild assumptions, that the IBMQ transmons suffer state leakage, with a $p$ value no larger than $5\times {10}^{-4}$ under a single-qubit operation. We also estimate the number of shots necessary for revealing leakage in a two-qubit system.

Figure 1

The spectra of nondefective quantum channels determines the maximum complexity of any time series that they generate, and the rank of the corresponding matrix of delayed vectors. Various homogenous, Markovian, completely positive and trace-preserving processes $\mathcal{E}\left(t\right)$ are illustrated. (a) The flow of the spectrum $\left\{{\lambda }_j^s\right\}$ of $T_{\mathcal{E}}^s$ is shown in the complex plane for $1\le s\le 2$ for a $d=2$ (i) unitary and (ii) nonunitary, and $d=3$ (iii) unitary and (iv) nonunitary process. Where $|{\lambda }_j|=1$, the flow is on the unit circle; where $|{\lambda }_j|<1$ there is an inexorable spiral towards the origin. All of these processes feature a stationary eigenvalue at 1 because of their trace-preserving nature. The time-varying expectation $\langle M\left(t\right)\rangle$, defined by a rank one initial state $\rho$, rank one measurement operator $M$, and $T_{\mathcal{E}}^t$, is shown in (b) and (c) for discrete instants $t=0,1,2...18$. Smooth curves show $\langle M\left(s\right)\rangle$ for continuous $0\le s\le 18$, and are a guide for the eye. Oscillation frequencies, phase offsets, decay envelopes, and vertical shifts may be traced back to the spectral properties of $T_{\mathcal{E}}$. Indeed, the different frequencies were also noted in the time series of classical dynamical systems [28]. The matrix of delayed vectors $V$, whose rank (in these nondefective cases) is no more than the number of distinct eigenvalues of $T_{\mathcal{E}}$, is shown in (d). If $U$ is a matrix drawn from the Haar ensemble and $Y=U^{1/3}$, unitary evolutions were constructed as $T_{\mathcal{E}}=Y\otimes \overline{Y}$ and nonunitary evolutions as $T_{\mathcal{E}}=0.08T_B+0.92(Y\otimes \overline{Y})$; $T_B$ is a random CPTP map generated via the algorithm in Ref. [35].

Figure 2

Circuits representing target (a) single-qubit and (b) two-qubit unitary evolutions. In each case the highlighted region shows a single step that is repeated $t$ times, where $t\in \{0,1,...2N-2\}$. Single-qubit gates are defined by three angles, and the two-qubit gate is a controlled-not.

Figure 3

The observed singular values for (a) single-qubit and (c) for two-qubit evolutions [shown in logarithmic scale for ease of comparison in the inset (d)]. The gray dots in inset (b) show the time series of an example evolution for a single qubit, with the continuous blue line showing a continuous time simulation of the target evolution. Thick gray bars correspond to the IBMQ device, while thin red bars correspond to a local simulation of the target evolution. The dashed line shows the tolerance level at a $p$ value of 0.001 [see Eq. (23)].

Figure 4

Number of validated singular values from a single qubit (a) 8192-shot ideal simulation (red bars) and (b) 8192-shot execution on IBMQ (gray bars). Two-qubit simulations and IBMQ data are shown in (c) and (d), respectively. Thin light green bars show 8192-shot ideal simulations of random unitary experiments in a (a) three-dimensional and (c) five-dimensional system. The thresholds from Eq. (23) were set at a $p$ value of 0.001. The dashed lines show an upper bound on the rank of a delayed vector matrix generated by a $d=2$ [in (a) and (b)] or $d=4$ [in (c) and (d)] dimensional system.

Figure 5

Number of validated singular values via a finite-shot ideal simulation of a four-dimensional system (thick red bars) and a five-dimensional system (thin light green bars) for different numbers of experimental shots. The thresholds from Eq. (23) were set at a $p$ value of 0.001. The multiplication factor for the number of shots used, compared to the main experiment which used 8192 shots, is shown in the lower left corner of each panel.