Independent state and measurement characterization for quantum computers

Correctly characterizing state preparation and measurement (SPAM) processes is a necessary step towards building reliable quantum processing units (QPUs). In this work, we discuss the subtleties behind separately measuring SPAM errors. We propose a protocol that can separately estimate SPAM errors, in the case where quantum gates are ideal. In the case where the quantum gates are imperfect, we derive bounds on the estimated SPAM error rates, based on gate error measures, which can be estimated independently of SPAM processes. Our method shows that the gauge ambiguity in characterizing SPAM operations can be resolved by assuming that there exists one qubit whose initial state is uncorrelated with other qubits in a QPU. We test the protocol on a publicly available five-qubit QPU and demonstrate its validity by comparing our results with simulations.

Figure 1

Circuits for determining the coefficient $s_{Z,t}$ on the target qubit $q_t$, assuming ideal gates. The one on the left/right gives ${\alpha }_a$ and ${\beta }_t$, respectively. Combinations of SPAM averaging gates are run as separate experiments (the outcome is flipped classically when the first $M$-averaging gate is $X$, indicated by an overhead circle). Adjacent single-qubit gates (grouped by the dashed box) are logically compiled to a single gate when running the circuits.

Figure 2

The circuit for estimating ${\beta }_{P,t}$ of an $N$-qubit system $q_t$, assuming ideal gates. The dotted box indicate the propagating cycle ${\mathcal{U}}_P$.

Figure 3

Top: Estimated single-qubit state preparation (in red) and measurement (in blue) error rates on ibmq-santiago QPU. Shaded regions represent the range of error rates consistent with the measured gate error, and error bars are $95\%$ CIs for the endpoints. Green dots indicate measured total SPAM error ${\epsilon }_{\text{SPAM}}$. All estimates are cutoff below $0\%$. Bottom: A simulation assuming $T_1,T_2$ relaxation gate errors on two-qubit gates and ideal one-qubit gates, using the device's specifications. Green stars mark the magnitudes of SP- and M-errors used in the simulation, which combine to ${\epsilon }_{\text{SPAM}}$ on each qubit. Darker regions mark the same bounds when all gate times are reduced to $1/5$ of their original values.

Figure 4

An expanded version of the two-qubit propagating circuits that are actually carried out. The restoring gates are given by ${\mathcal{D}}_r^{†}=\mathcal{C}{\mathcal{D}}^{†}{\mathcal{C}}^{†}$. Note that SPAM averaging gates are compiled with adjacent dressing gates from randomized compiling (denoted by dashed boxes) and are implemented as one gate, hence there is only one noise channel ${\mathcal{E}}_1^{\otimes 2}$. The red dashed line indicates the place where we make the comparison (see text).