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arXiv:1912.13131 · Physical Review Letters

Eliminating Leakage Errors in Hyperfine Qubits

David Hayes, Daniel T. Stack, Bryce Bjork, Andrew C. Potter, Charles H. Baldwin, Russell Stutz

Published Apr 27, 2020• 31 citations
arXivHTMLPDF

Abstract

Population leakage outside the qubit subspace presents a particularly harmful source of error that cannot be handled by standard error correction methods. Using a trapped $^{171}{\mathrm{Yb}}^{+}$ ion, we demonstrate an optical pumping scheme to suppress leakage errors in atomic hyperfine qubits. The selection rules and narrow linewidth of a quadrupole transition are used to selectively pump population out of leakage states and back into the qubit subspace. Each pumping cycle reduces the leakage population by a factor of $\ensuremath{\sim}3$, allowing for an exponential suppression in the number of cycles. We use interleaved randomized benchmarking on the qubit subspace to show that this pumping procedure has negligible side effects on the qubit subspace, bounding the induced qubit memory error by $\ensuremath{\le}2.0(8)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}\text{ }\text{ }\mathrm{per}$ cycle, and qubit population decay to $\ensuremath{\le}1.4(3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}\text{ }\text{ }\mathrm{per}$ cycle. These results clear a major obstacle for implementations of quantum error correction and error mitigation protocols.

DOI
10.1103/physrevlett.124.170501

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Population leakage outside the qubit subspace presents a particularly harmful source of error that cannot be handled by standard error correction methods. Using a trapped 171Yb+^{171}{\mathrm{Yb}}^{+}171Yb+ ion, we demonstrate an optical pumping scheme to suppress leakage errors in atomic hyperfine qubits. The selection rules and narrow linewidth of a quadrupole transition are used to selectively pump population out of leakage states and back into the qubit subspace. Each pumping cycle reduces the leakage population by a factor of \ensuremath∼3\ensuremath{\sim}3\ensuremath∼3, allowing for an exponential suppression in the number of cycles. We use interleaved randomized benchmarking on the qubit subspace to show that this pumping procedure has negligible side effects on the qubit subspace, bounding the induced qubit memory error by \ensuremath≤2.0(8)\ifmmode×\else\texttimes\fi10\ensuremath−5  per\ensuremath{\le}2.0(8)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}\text{ }\text{ }\mathrm{per}\ensuremath≤2.0(8)\ifmmode×\else\texttimes\fi10\ensuremath−5  per cycle, and qubit population decay to \ensuremath≤1.4(3)\ifmmode×\else\texttimes\fi10\ensuremath−7  per\ensuremath{\le}1.4(3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}\text{ }\text{ }\mathrm{per}\ensuremath≤1.4(3)\ifmmode×\else\texttimes\fi10\ensuremath−7  per cycle. These results clear a major obstacle for implementations of quantum error correction and error mitigation protocols.
Leakage errors take qubits out of the computational subspace and will accumulate if not addressed. A leaked qubit will reduce the effectiveness of quantum error correction protocols due to the cost of implementing leakage reduction circuits and the harm caused by interacting leaked states with qubit states. Ion trap qubits driven by Raman gates have a natural choice between qubits encoded in magnetically insensitive hyperfine states that can leak and qubits encoded in magnetically sensitive Zeeman states of the electron spin that cannot leak. In our previous work, we compared these two qubits in the context of the toric code with a depolarizing leakage error model and found that for magnetic-field noise with a standard deviation less than 32 $\ensuremath{\mu}\mathrm{G}$ that the 174Yb+^{174}\mathrm{Yb}^{+}174Yb+ Zeeman qubit outperforms the 171Yb+^{171}\mathrm{Yb}^{+}171Yb+ hyperfine qubit. Here we examine a physically motivated leakage error model based on ions interacting via the M\o{}lmer-S\o{}renson gate. We find that this greatly improves the performance of hyperfine qubits but the Zeeman qubits are more effective for magnetic-field noise with a standard deviation less than 10 $\ensuremath{\mu}\mathrm{G}$. At these low magnetic fields, we find that the best choice is a mixed qubit scheme where the hyperfine qubits are the syndrome qubits and the leakage is handled without the need of an additional leakage reduction circuit.
Leakage is a quantum error process in which a qubit ends up in a state outside of the computational subspace. In trapped-ion hyperfine qubits, leakage can occur as a result of spontaneous emission from an excited state during a two-qubit gate. This error can be detected by a circuit gadget that couples a data qubit to an ancilla qubit, followed by a measurement of the ancilla. The effects of leakage errors on the circuit can then be removed by post-selecting shots in which leakage was not detected. We first experimentally demonstrate improved two-qubit gate fidelity as measured by randomized benchmarking and cycle benchmarking. We then benchmark the use of leakage detection on system-level mirror benchmarking circuits. Finally, we present a protocol to measure the failure rate of the leakage detection gadget, defined as the probability that the gadget produces an error on the data qubit while not heralding the ancilla.
Many systems used for quantum computing possess additional states beyond those defining the qubit. Leakage out of the qubit subspace must be considered when designing quantum error correction codes. Here we consider trapped ion qubits manipulated by Raman transitions. Zeeman qubits do not suffer from leakage errors but are sensitive to magnetic fields to first order. Hyperfine qubits can be encoded in clock states that are insensitive to magnetic fields to first order, but spontaneous scattering during the Raman transition can lead to leakage. Here we compare a Zeeman qubit (174Yb+^{174}\mathrm{Yb}^{+}174Yb+) to a hyperfine qubit (171Yb+^{171}\mathrm{Yb}^{+}171Yb+) in the context of the surface code. We find that the number of physical qubits required to reach a specific logical qubit error can be reduced by using 174Yb+^{174}\mathrm{Yb}^{+}174Yb+ if the magnetic field can be stabilized with fluctuations smaller than $10\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{G}$.
Abstract The leakage of quantum information out of the two computational states of a qubit into other energy states represents a major challenge for quantum error correction. During the operation of an error-corrected algorithm, leakage builds over time and spreads through multi-qubit interactions. This leads to correlated errors that degrade the exponential suppression of the logical error with scale, thus challenging the feasibility of quantum error correction as a path towards fault-tolerant quantum computation. Here, we demonstrate a distance-3 surface code and distance-21 bit-flip code on a quantum processor for which leakage is removed from all qubits in each cycle. This shortens the lifetime of leakage and curtails its ability to spread and induce correlated errors. We report a tenfold reduction in the steady-state leakage population of the data qubits encoding the logical state and an average leakage population of less than 1 × 10 −3 throughout the entire device. Our leakage removal process efficiently returns the system back to the computational basis. Adding it to a code circuit would prevent leakage from inducing correlated error across cycles. With this demonstration that leakage can be contained, we have resolved a key challenge for practical quantum error correction at scale.
Leakage errors occur when a quantum system leaves the two-level qubit subspace. Reducing these errors is critically important for quantum error correction to be viable. To quantify leakage errors, we use randomized benchmarking in conjunction with measurement of the leakage population. We characterize single qubit gates in a superconducting qubit, and by refining our use of derivative reduction by adiabatic gate pulse shaping along with detuning of the pulses, we obtain gate errors consistently below 10^{-3} and leakage rates at the 10^{-5} level. With the control optimized, we find that a significant portion of the remaining leakage is due to incoherent heating of the qubit.
Many realizations of solid-state qubits involve couplings to leakage states lying outside the computational subspace, posing a threat to high-fidelity quantum gate operations. Mitigating leakage errors is especially challenging when the coupling strength is unknown, e.g., when it is caused by noise. Here we show that simple pulse sequences can be used to strongly suppress leakage errors for a qubit embedded in a three-level system. As an example, we apply our scheme to the recently proposed charge quadrupole (CQ) qubit for quantum dots. These results provide a solution to a key challenge for fault-tolerant quantum computing with solid-state elements.