Maximizing the nondemolition nature of a quantum measurement via an adaptive readout protocol

TL;DR

This work tackles the challenge of non-ideal QND readout in quantum error correction by introducing an adaptive readout protocol for an $D=8$ nuclear qudit that switches to the dark-state subspace after a single positive outcome, leveraging negative-result measurements to minimize backaction. The method is grounded in a tripartite Hamiltonian $H=H_S+H_A+H_C$ where non-commuting terms induce both $\Delta m=\pm1$ and $\Delta m=\pm2$ transitions, and is implemented on $^{123}$Sb in silicon with an electron ancilla and SET readout. Empirically, the adaptive readout increases the average readout fidelity from $98.93\pm0.07\%$ to $99.61\pm0.04\%$ and achieves a roughly threefold speedup, while simulations show robustness to imperfect ancilla readout. Extending the analysis to $^{73}$Ge with Pauli spin blockade demonstrates that measurement-induced backaction is a general issue across platforms, underscoring the broad relevance of the adaptive scheme for high-fidelity, fast quantum syndrome extraction in fault-tolerant architectures.

Abstract

Quantum error correction (QEC) requires non-invasive measurements for fault tolerant quantum computing. Deviations from ideal quantum non-demolition (QND) measurements can disturb the encoded information. To address this challenge, we develop a readout protocol for a $D-$dimensional system that, after a single positive outcome, switches to probing only the $D{-}1$ remaining subspace. This adaptive switching strategy minimizes measurement-induced errors by relying on negative-result measurement results that do not perturb the Hamiltonian. We apply the protocol on an 8-dimensional $^{123}{\rm Sb}$ nuclear qudit in silicon, and achieve an increase in the readout fidelity from $(98.93\pm0.07)\%$ to $(99.61\pm0.04)\%$, while reducing threefold the overall readout time. To highlight the broader relevance of measurement-induced errors, we study a 10-dimensional $^{73}{\rm Ge}$ nuclear spin read out through Pauli spin blockade, revealing nuclear spin flips arising from hyperfine and quadrupole interactions. These results unveil the effect of non-ideal QND readout across diverse platforms, and introduce an efficient readout protocol that can be implemented with minimal FPGA logic on existing hardware.

Figures (5)

• Figure 1: Deviation from ideal QND measurements due to an eigenbasis mismatch of coupled and decoupled systems. a) In the coupled system, the qubit or qudit is coupled to an ancilla, which is tunnel coupled to a reservoir or quantum dot (QD). b) If the ancilla tunnels onto the reservoir/QD, the coupling Hamiltonian $H_c$ reduces to 0. This process induces spin flips if $[H_S, H_C]\neq0$. c) One QND cycle consists of coupling, conditional Electron Spin Resonance (ESR), and decoupling the ancilla. d) The eigenstates of the coupled Hamiltonian do not fully overlap with the tensor basis due to the quadrupole and hyperfine interaction. The quadrupole shows up as $\Delta m=\pm1, 2$ off-diagonal elements in this matrix, whereas the hyperfine enables coupling between nuclear spin $\Delta m=\pm1$ and electron spin $\Delta m_e=\mp 1$ components, visible as off-diagonal elements in the top left and bottom right quadrants. e) In the decoupled case, the hyperfine interaction reduces to zero ($H_C=0$) and the influence of the quadruople becomes more pronounced. f) The overlap of decoupled and coupled eigenstates - $|{m^\emptyset}\rangle$ and $|m^{\updownarrow}\rangle$ respectively - when the ancilla spin state is $\ket{\downarrow}$ (light green) or $\ket{\uparrow}$ (dark green).
• Figure 2: Characterization of measurement-induced nuclear spin flips in $\mathbf{^{123}\rm Sb}$a) Pulse sequence to probe measurement-induced dynamics. We sequentially perform QND readout of the nuclear states $\ket{-7/2}$ through $\ket{+7/2}$, repeated $N$ times. The nuclear state is detected by a short blip of current, caused by the ancillary electron tunneling to a nearby SET. If the blip exceeds the threshold, we count it as a readout event. b) Raw readout results. We determine the most likely state (red line) by window filtering the raw data, to exclude dark counts. We observe the nucleus jumping by either $\Delta m=\pm2$ (purple) or $\Delta m =\pm1$ (green). c) Extracted transition probability matrix. The first and second off-diagonal entries (green and purple) correspond to $\Delta m=1$ and $\Delta m =2$ jumps respectively. d) Simulated transition probability matrix based on eigenstate overlap of the coupled and decoupled nucleus. See Supplemental Materials Section I for model details.
• Figure 3: Readout improvement using an adaptive QND protocol. a) Decision tree of conventional repeated QND readout (RR) and corresponding pulse sequence. b) Decision tree of the adaptive QND readout (AR) protocol and an example of a corresponding pulse sequence. As soon as a readout event is detected, the protocol switches to measuring the dark state space $N$ times. c) Nuclear flip probabilities during RR in the best-case scenario. d) Nuclear flip probabilities of AR in the best-case scenario. The adaptive protocol reduces $\Delta m=1$ (green) and $\Delta m=2$ (purple) transition probabilities. e) Comparison of AR (blue) and RR (orange), simulated (lines) and data (dots). The AR data approaches the ionization shock limit for one shot. f) Average readout fidelity comparison vs. number of readout shots and vs. time for RR (orange, bottom axis) and AR (blue, top axis). The black dashed line indicates the simulated RR fidelity if the readout is solely limited by ionization shock (i.e. in the limit of perfect ancilla readout).
• Figure 4: Simulated comparison of repeated and adaptive QND readout protocols.a) Optimal number of repeated readout shots (red line) increases as ancilla fidelity decreases. b) Maximum fidelity in repeated readout $F_{\rm RR}$ decreases with shot number $N_{\rm RR}$ due to non-ideal QND conditions. c) Adaptive readout achieves higher fidelities overall compared to repeated readout. When the ancilla readout fidelity approaches 1, the AR readout fidelity $F_{\rm AR}$ becomes insensitive to the number of shots $N_{\rm AR}$ due to the lack of tunneling events. The simulated optimal number of shots in that regime (red line) is subject to numerical fluctuations. d) Readout fidelity in adaptive readout remains robust against increasing dark space shots $N_{\rm AR}$, confirming suppression of measurement-induced errors. We ascribe oscillations in even-odd number of shots to the definition of the threshold ${\rm th =} \lceil N_{\rm AR}/2\rceil$, which slightly favors even $N_{\rm AR}$.
• Figure 5: Measurement-induced spin flips in a $\mathbf{^{73}\rm Ge}$ donor coupled to a double quantum dot.a) Schematic of the system. b) Measurement-induced nuclear spin transition matrix for one ESR spectroscopy frequency sweep of 201 points. The transition matrix shows primarily $\Delta m =1$ (green) and $\Delta m = 2$ (purple) jumps, similar to backaction dynamics seen in the $^{123}\rm Sb$ system. c--d) Estimated transition probabilities per QND cycle. Increasing the magnetic field from $B_0 = 0.3T$ (light) to $B_0 = 1T$ (dark) reduces the transition probabilities.