Real-time measurement error mitigation for one-way quantum computation - Nature

Scientific Reports (2026) Cite this article We’re sharing this article early to provide faster access to peer-reviewed, accepted research. It is citable and carries a permanent DOI. This version is subject to further edits and will be replaced automatically by the final Version of Record. All legal disclaimers apply. We propose a quantum error mitigation scheme for single-qubit measurement errors, particularly suited for one-way quantum computation. Contrary to well established error mitigation methods for circuit-based quantum computation that require the circuits to be run several times, our method is capable of mitigating measurement errors in real time during the processing measurements in the one-way computation. For that, an ancillary qubit register is entangled with the to-be-measured qubit and additionally measured afterwards. By using a voting protocol on all measurement outcomes, occurring measurement errors can be mitigated in real time while the one-way computation continues. We provide an analytical expression for the probability to detect a measurement error depending on the error rate and the number of ancilla qubits. From this, we derive an estimate of the ancilla register size for a given measurement error rate and a required success probability to detect a measurement error. Additionally, we consider the CNOT gate error in our mitigation method and investigate how this influences the probability to detect a measurement error. Finally, we show in proof-of-principle simulations and in experiments executed on IBM quantum hardware that our method is capable of reducing the measurement errors significantly in a one-way quantum computation with only a small number of ancilla qubits.We thank Ferdinand Schmidt-Kaler and his group for helpful discussions and the provided hardware data of measurement error rates. This work is funded by the European Union’s Horizon Europe Framework Programme (HORIZON) under the ERA Chair scheme with grant agreement no. 101087126 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 429529648 – TRR 306 QuCoLiMa (“Quantum Cooperativity of Light and Matter”). This work is supported with funds from the Ministry of Science, Research and Culture of the State of Brandenburg within the Centre for Quantum Technologies and Applications (CQTA).Open Access funding enabled and organized by Projekt DEAL. This work is funded by the European Union’s Horizon Europe Framework Programme (HORIZON) under the ERA Chair scheme with grant agreement no. 101087126 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 429529648 – TRR 306 QuCoLiMa (“Quantum Cooperativity of Light and Matter”). This work is supported with funds from the Ministry of Science, Research and Culture of the State of Brandenburg within the Centre for Quantum Technologies and Applications (CQTA).Tobias Hartung and Stephan Schuster contributed equally to this work.Computing and Information Systems, Northeastern University-London, Devon House, St Katharine Docks, London, E1W 1LP, UKTobias HartungKhoury College of Computer Sciences, Northeastern University, #202, West Village Residence Complex H, 440 Huntington Ave, Boston, MA, 02115, USATobias HartungQuantum Optics and Quantum Information Group, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstr. 1, 91058, Erlangen, GermanyStephan Schuster & Joachim von ZanthierComputation-Based Science and Technology Research Center, The Cyprus Institute, 20 Kavafi Street, 2121, Nicosia, CyprusKarl JansenDeutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738, Zeuthen, GermanyKarl JansenSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarCorrespondence to Karl Jansen.The authors declare no competing interests.Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.The dependence of N on \(\varepsilon \) is shown in Eq. (4) in the main text. In the following, a detailed derivation of this equation is presented.The regularized incomplete beta function from Eq. (2), modelling the misidentification probability, can be written in the polynomial approximation shown in Eq. (3) as:Assuming that N is an odd number, since the voting protocol would be indefinite otherwise, we can write the floor function \(\lfloor N/2 \rfloor \) as:The binomial coefficient in Eq. (A1) can thus be simplified as follows, using Stirling’s approximation for factorials \(n! \approx \sqrt{2\pi n}(n/e)^n\):This yields a compact approximation for the misidentification probability:In order to invert this function \(\varepsilon (N)\) for a given fixed \(r_{\lnot b, b}\) and thus find \(N(\varepsilon )\) we can use the Lambert W-function. This function is defined as the inverse of \(ye^y\),First, we will rewrite Eq. (A2) in the formwith \(R=4r_{\lnot b, b}\). Next, we apply \(R^x=e^{x\ln (R)}\) and find the required form to use the Lambert W-functionNote that the argument of the Lambert W(z) has to fulfill \(z\ge -1/e\). Since \(z=-1/e\) is the global minimum of \(z=ye^y\). This is fulfilled for any \(\varepsilon \) if \(r_{\lnot b, b}\le 0.25\),For higher measurement errors, the polynomial approximation of the regularized incomplete beta function deviates significantly from the exact function. In this case, other methods to extract the required N for a given \(\varepsilon \) can be employed, e.g., iteratively calculating \(\varepsilon \) from \(I_{r_{\lnot b, b}}(N-\lfloor N/2\rfloor ,1+\lfloor N/2\rfloor )\) while increasing N.As already discussed in the main text, Fig. 5 and Fig. 6 show three regimes for \(\varepsilon _{\text {est}}\) in dependence of the \(\textrm{CNOT}\) error \(\gamma \):Immediate improvement when we increase NInitial worsening, then improvement when we increase NNo improvement at all when we increase NIn this section, we will discuss the different regimes in more detail. Fig. 10ashows the change in the misidentification probability \(\varepsilon \) from \(N=1\) (no verification register) to \(N=3\) in dependence of the initial error rate \(m_{\lnot b, b}\) and the \(\textrm{CNOT}\) error \(\gamma \) (cf. Eq. (9)). In the case that \(\gamma \) is too large, N needs to be sufficiently large as we will not see an initial improvement for small N. The critical threshold \(\gamma _{\text {crit}}\) which separates initial improvement \(\varepsilon _{N=3}-\varepsilon _{N=1}0\) is shown in Fig. 10bas a function of the initial error rate \(m_{\lnot b, b}\). If the \(\textrm{CNOT}\) error \(\gamma \) is less than \(\gamma _{\text {crit}}\), we are in the region of immediate improvement and otherwise we get an initial worsening of the misidentification probability.(a) Heatmap of the change in \(\varepsilon \) from \(N=1\) to \(N=3\), \(\varepsilon _{N=3}-\varepsilon _{N=1}\), in dependence of the initial error rate \(m_{\lnot b, b}\) and the \(\textrm{CNOT}\) error \(\gamma \). (b) The solid blue lines show the critical threshold \(\gamma _{\text {crit}}\) which separates initial improvement \(\varepsilon _{N=3}-\varepsilon _{N=1}0\) in (a) as a function of the initial error rate \(m_{\lnot b, b}\). The dotted black line shows \(\gamma _{\text {crit}}=m_{\lnot b, b}\).To answer the question of which N gives the lowest misidentification probability \(\varepsilon \) for a given initial error \(m_{\lnot b, b}\) and a given \(\textrm{CNOT}\) error \(\gamma \), we calculated this best N iteratively from Eq. (9) and plotted it as a function of \(m_{\lnot b, b}\) and \(\gamma \) in Fig. 11. The plot shows that for realistic \(\textrm{CNOT}\) errors \(\gamma <5\%\), a bigger N is (within reasonable limits) better once the initial worsening is overcome for any initial errors \(m_{\lnot b, b}\le 0.2\). If the \(\textrm{CNOT}\) error is higher than \(5\%\), there exists a transition region depending on the initial error rate \(m_{\lnot b, b}\) for which there is an optimal \(1