Robust characterization of leakage errors

Abstract

Leakage errors arise when the quantum state leaks out of some subspace of interest, for example, the two-level subspace of a multi-level system defining a computational 'qubit', the logical code space of a quantum error-correcting code, or a decoherence-free subspace. Leakage errors pose a distinct challenge to quantum control relative to the more well-studied decoherence errors and can be a limiting factor to achieving fault-tolerant quantum computation. Here we present a scalable and robust randomized benchmarking protocol for quickly estimating the leakage rate due to an arbitrary Markovian noise process on a larger system. We illustrate the reliability of the protocol through numerical simulations.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. Defining leakage ratesMany experimental implementations of logical d 1 -level qudits (typically = d 2 1 , giving a qubit) are embedded in a d-level quantum system by taking only the first d 1 levels ñ ¼ ñ | |d 1 , , 1 . Formally, we can consider a OPEN ACCESS RECEIVED

Errors and pseudothresholds for incoherent and coherent noise

“…In general, open-loop control techniques can be used to transform error channels that have a larger coherent character into errors that have less coherent character [41,42]. At the gate level the addition of Pauli twirling gates [43,44] can also reduce the coherent noise. Given the negative effect of coherent errors on the pseudothreshold, we expect these coherent noise reducing methods will be essential for achieving logical qubits that outperform physical qubits.…”

Handling leakage with subsystem codes

“…We have quantified several crossover points below which these correlated errors are more damaging than the entropic effects that are well-handled by the subspace surface code. These error rates range between approximately 10 and 10 5 3 -at the low distances we have considered, and may be relevant to limiting error models in certain architectures, such as spontaneous scattering in ion traps.…”

“…Leakage is a particularly damaging error that occurs when a qubit excites to a state outside of its two-level computational subspace [3][4][5][6][7][8][9][10]. Much work has been devoted to characterizing and reducing leakage at the experimental level in different qubit architectures, including photonics [11], quantum dots [12][13][14][15][16][17], superconductors [18][19][20][21][22][23][24][25][26][27][28], topological qubits [29], and ion traps [30][31][32].…”

Subspace benchmarking high-fidelity entangling operations with trapped ions

“…A similar problem arises in single-qubit RB if there are errors that cause population to leave the computational subspace, called leakage errors. Previously, several protocols to measure leakage rates were proposed, which we refer to as leakage benchmarking (LB) [18][19][20]. We now show LB can be performed along with SRB-lite to create a more complete picture of the errors in the MS gate.…”

“…The basic idea behind our procedure is to isolate a subspace of the total system to perform RB, hence we call * charles.baldwin@honeywell.com the method subspace randomized benchmarking (SRB). We then use leakage benchmarking [18][19][20] to identify errors that may drive population outside of this subspace. We show that these different methods can be combined into a single procedure and each effect can be individually measured.…”