Requires that the noise can be scaled by a controlled factor and that the observable's dependence on noise strength is captured by the chosen fit. The extrapolation model is an assumption, and a wrong model produces a confidently wrong number. It reduces bias while amplifying variance, so the shot cost rises, and it returns an expectation value that cannot be handed to a coherent downstream subroutine. Temme, Bravyi and Gambetta state that the size of the circuits to which these techniques can be applied is limited by the rate at which errors are introduced.

A circuit, a target observable, a noisy device, a shot budget, and — for the model-based methods — a learned characterization of the device noise.

1. Noisy expectation value → Bias-reduced expectation valuerun at scaled noise, extrapolate to zero

   The mitigated estimate is built as $\hat E_K^n(\lambda) = \sum_{j=0}^n \gamma_j \hat E_K(c_j\lambda)$, a linear combination of estimates measured at $n+1$ rescaled noise strengths $c_j\lambda$ ($c_0=1$), obtained physically by replacing the drive $J_\alpha(t)$ with $J_\alpha^j(t) = c_j^{-1}J_\alpha(c_j^{-1}t)$ run for time $T_j = c_jT$, which reproduces the state noise $c_j\lambda$ would produce at the original $T$. The weights solve $\sum_j \gamma_j = 1$ and $\sum_j \gamma_j c_j^k = 0$ for $k=1,\ldots,n$, giving $\gamma_j = \prod_{m\neq j} c_m(c_j-c_m)^{-1}$, approximation: this cancels the noise series $E_K(\lambda) = E^* + \sum_{k=1}^n a_k\lambda^k + R_{n+1}$ only through order $\lambda^n$; the remainder obeys $|R_{n+1}| \le \|A\|\,l_{n+1}(\lambda T)^{n+1}/(n+1)!$, so the error against the true value is bounded by $\Gamma_n\big(\delta^* + \|A\|\,l_{n+1}(\lambda T)^{n+1}/(n+1)!\big)$, with $\Gamma_n=\sum_j|\gamma_j|c_j^{n+1}$ and $\delta^*=\max_j|\delta_j|$ the largest sampling error, amplifying both the remainder and the sampling error. assumption: the generator $\mathcal L$ in $\partial_t\rho=-i[K(t),\rho]+\lambda\mathcal L(\rho)$ stays invariant under this time rescaling and independent of $J_\alpha(t)$, with $\lambda\ll1$; more precisely, the expansion needs $NT\lambda$ small ($N$ qubits, $T$ evolution time), not $\lambda$ alone.

   approximationassumption

A bias-reduced expectation-value estimate, with a variance — equivalently a sampling overhead — that grows with circuit volume.

given  a circuit C, an observable O, an extrapolation model,
       and noise scale factors s_1 < s_2 < ... < s_m  (s_1 = 1, the device as it is)

for each scale factor s:
    build C_s -- the same circuit with the device noise amplified by s
    measure <O> on C_s over enough shots for the target variance

fit the chosen model to the points (s, <O>_s)
    # Richardson's deferred approach to the limit, or another model

return the fit evaluated at s = 0

# the extrapolation model is an assumption, and a wrong model produces a
# confidently wrong number
# bias falls while variance rises, so the shot cost goes up
# what comes back is an expectation value: it cannot be handed to a coherent
# downstream subroutine

Temme, Bravyi and Gambetta's own accounting, their Eq. (6): the protocol runs the circuit at $n+1$ rescaled noise levels, and the deviation from the exact expectation value is bounded by two terms — the largest experimental error $\delta^*$ and a truncation term suppressed factorially as $(\lambda T)^{n+1}/(n+1)!$ up to the paper's stated prefactors — both amplified by $\Gamma_n = \sum_{j=0}^{n} |\gamma_j| c_j^{n+1}$. Extrapolation buys lower bias at the price of amplified error and shot cost, with no extra qubits. Stated in the full text; the abstract carries no formula.

The headline demonstration — the 127-qubit IBM Eagle kicked-Ising experiment reported as evidence for the utility of quantum computing before fault tolerance (Kim et al., Nature 618, 500 (2023)) — was afterwards reproduced classically and in places exceeded: by belief-propagation tensor networks (Tindall, Fishman, Stoudenmire, Sels), whose results were more accurate and more precise than the processor's, and by sparse Pauli dynamics running on a single laptop core orders of magnitude faster than the reported quantum walltime (Begusic, Chan). The mitigation itself worked; the quantum-advantage reading did not survive.

• Four-qubit Richardson-extrapolation numerics of Temme, Bravyi and Gambetta

  • Error mitigation for short-depth quantum circuits

    Kristan Temme, Sergey Bravyi, Jay M. Gambetta · 2016

  About

  The demonstration carried in the paper that introduces the method, from IBM T. J. Watson Research Center; published as Phys. Rev. Lett. 119, 180509 (2017). The question it puts to itself is narrow: given a short-depth circuit that can be re-run at deliberately rescaled noise rates, how far does Richardson's deferred approach to the limit push the estimate of the noiseless expectation value $E^*$ before the sampling error takes over. The paper calls the three plots "three numerical examples", and they are the only place in it where the extrapolation is actually run.

  Methods

  The circuit is run $n+1$ times at rescaled noise rates $\lambda_j = c_j\lambda$ and the estimates are combined with the Richardson coefficients of Eqs. (3)-(4). The $n+1$ noise levels follow the paper's own rescaling protocol, Eq. (5): the drive couplings are replaced by $J_\alpha^j(t) = c_j^{-1}J_\alpha(c_j^{-1}t)$ and the evolution is run for the stretched time $T_j = c_jT$, which the paper proves leaves the same state as evolving at rate $c_j\lambda$ directly, $\rho^j_\lambda(T_j) = \rho_{c_j\lambda}(T)$. The paper does not say which of the two the Fig. 1 numerics used. That equivalence carries a side condition the paper states three times and which the method inherits: the noise generator $\mathcal{L}$ is not specified in form, but is required to be invariant under time rescaling and independent of the couplings $J_\alpha(t)$ in $K(t)$ (main text, model setup), and in Supplementary II the map $\rho'_\lambda(T') = \rho_{c\lambda}(T)$ is claimed only if $\mathcal{L}$ does not depend on the couplings and is constant in time; the Conclusions add that the scheme \"hinges on the assumption of a large time-scale separation between the dominant noise and the controlled dynamics\". Orders $n = 0, 1, 2, 3$ are plotted, $n = 0$ being the uncorrected measurement, and the $\{c_j\}$ are drawn as a random partition of the interval $[1,4]$.

  Data

  Synthetic, generated inside the paper: a control problem of $d = 6$ drift steps on $N = 4$ system qubits, each step of time $t = 2$. One drift step evolves under $K_R(t) = U_N(\theta)K_0U_N^{\dagger}(\theta)$, where the single-qubit product unitary $U_N(\theta) \in SU(2)^{\otimes N}$ is Haar-random and the drift Hamiltonian $K_0 = \sum_{i,j}J_{i,j}X_iZ_j$ is taken over a random graph with Gaussian-distributed couplings $J_{i,j}$. After the full evolution $T = td$ a randomly chosen multi-qubit Pauli operator is measured. There is no external dataset and no molecule.

  Code

  None found yet.

  Results

  Classical numerical simulation only: no device is involved and the paper names no simulation package. Fig. 1 reports the error $\Delta E = |E^* - \hat{E}^n_K(\lambda)|$ under three noise models -- (a) single-qubit depolarizing, (b) amplitude damping together with dephasing at ratio $\lambda_1/\lambda_2 = 1.5$, (c) a highly non-Markovian setting in which each qubit is coupled to its own single-qubit bath. All three panels are plotted at the same perturbation measured in the depolarizing strength $\epsilon = 10^{-3}\ldots10^{-2}$, the noise parameter being set to $\lambda = -1/2\log(1-\epsilon)$. At the bottom of that range, $\epsilon \sim 10^{-3}$, the paper states the relative error can be reduced to $\Delta E \sim 10^{-6} - 10^{-11}$, and states in the same breath that the precision is then essentially determined by the sampling error $\delta^*$, which was neglected in the plots. The figure therefore measures the truncation term of the method, not its shot cost.

• Pulse-stretched zero-noise extrapolation on IBM's five-qubit transmon processor

  • Extending the computational reach of a noisy superconducting quantum processor

    Abhinav Kandala, Kristan Temme, Antonio D. Corcoles, Antonio Mezzacapo, Jerry M. Chow, Jay M. Gambetta · 2018

  About

  A hardware realisation of exactly the Temme-Bravyi-Gambetta rescaling protocol -- the noise amplified by stretching the control pulses -- by the same IBM group (Kandala, Temme, Corcoles, Mezzacapo, Chow, Gambetta); arXiv lists it as published in Nature 567, 491 (2019). The problem it sets is the one Temme et al. left open: the rescaling trick asks for the noise rate to be amplified by a known factor, and this paper's stated primary objective is to obtain, experimentally, improved estimates of the noiseless expectation value $E^*$ despite using noisy quantum hardware. Its abstract states separately that the protocol is demonstrated with no additional hardware modifications.

  Methods

  Noise is amplified by stretching the control pulses: the paper shows that if the noise is time-translation invariant, a measurement of the expectation value after evolution under a scaled drive for a time $c_iT$ -- within the coherence window -- is equivalent to a measurement under an amplified noise strength $c_i\lambda$. For single-qubit gates the lengths of the $4\sigma$ Gaussian pulses and the buffer times are stretched by the desired $c_i$ and re-calibrated; for the two-qubit cross-resonance gate the pulse lengths, rise-fall times and buffer times are all stretched and re-calibrated to a $ZX_{\pi/2}$ gate, with the CR drive deliberately kept in a low-power regime where the strengths of the interaction terms in the drive scale linearly with drive amplitude, because a non-linear regime gives interaction strengths that do not scale appropriately and unphysical mitigated expectation values. Estimates are then combined by Richardson extrapolation, Eq. (2). Three experiments: identity-equivalent random single-qubit Clifford sequences at $c = 1, 2, 3, 4$ with $10^5$ samples, extrapolated to first, second and third order; identity-equivalent two-qubit Clifford sequences on a Bell state at $c = 1, 1.5$ with a first-order extrapolation; and a hardware-efficient variational eigensolver driven by simultaneous perturbation stochastic approximation (SPSA). The eigensolver's stretch factors are stated per configuration, not once for the paper: for the $J/B = 1$ magnetism run of Fig. 3, each iteration measures the trial-state energies at $c = 1, 1.5$ with $10^4$ samples, mitigates them by first-order Richardson and hands them to the optimizer, the controls of the final 25 iterations are averaged for the final state measurement, and the final energy comes from a linear extrapolation over $c = 1, 1.1, 1.25, 1.5$ with $10^5$ samples; the chemistry runs of Fig. 4 use their own final stretch factors, given in the panel legends. Error bars on the mitigated estimates come from bootstrapping the experimental data set rather than from repeated runs.

  Data

  Two problem families, both stated in the paper. First, an anti-ferromagnetic four-qubit Heisenberg model on a square lattice in an external field, $H = J\sum_{\langle ij\rangle}(X_iX_j + Y_iY_j + Z_iZ_j) + B\sum_i Z_i$, mapped over a range of $J/B$. Second, the electronic structure of H2 and LiH, mapped onto two and four qubits respectively using fermionic spin-parity symmetries and the freezing of core-shell orbitals, with the details deferred to Kandala et al., Nature 549, 242 (2017); this paper does not restate the basis set.

  Code

  None found yet.

  Results

  Five fixed-frequency transmon qubits and superconducting coplanar-waveguide resonators for coupling, control and readout, all fabricated on a Si wafer, qubit frequencies $\omega_{01}/2\pi \sim 5$-$5.3$ GHz, typical $T_1$ and $T_2$ in the 40-70 $\mu$s range and typical readout errors $\epsilon_r < 0.05$ at 2 $\mu$s integration. For the four-qubit Heisenberg model at $J/B = 1$, mitigation changes what circuit depth buys: at $c_i = 1$ the energy error improves over that of the lowest-energy separable state only up to depth $d = 2$, while the mitigated estimates keep improving up to $d = 5$ trial states -- and the $d = 5$ optimization at $J/B = 1$ runs over 68 variational parameters simultaneously. For chemistry, Fig. 4 reports mitigated energies in Hartree against a range of interatomic distances for H2 at trial-circuit depth $d = 2$ and LiH at $d = 3$, with the panel legends giving the final stretch factors as $c = 1, 1.25, 1.5, 2$ for H2 and $c = 1, 1.1, 1.25, 1.5$ for LiH. The paper reports its own ceiling back in Fig. 2, the Clifford-sequence figure: the eventual decay of the mitigated curves there shows that the method cannot be applied indefinitely and is ultimately limited by the quantum coherence of the device.

• Probabilistic-error-amplified ZNE on the 127-qubit Eagle processor ibm_kyiv

  • Evidence for the utility of quantum computing before fault tolerance

    Youngseok Kim, Andrew Eddins, Sajant Anand, Ken Xuan Wei, Ewout van den Berg, Sami Rosenblatt, Hasan Nayfeh, Yantao Wu, Michael Zaletel, Kristan Temme, Abhinav Kandala · 2023

  About

  The IBM Quantum experiment, with collaborators at UC Berkeley, Lawrence Berkeley National Laboratory and RIKEN iTHEMS, that reported expectation values from a 127-qubit processor at a circuit volume the authors argue lies beyond brute-force classical simulation; published as Nature 618, 500-505 (2023). It is the demonstration the method's contested note refers to. The paper is explicit that it chose ZNE over probabilistic error cancellation because PEC's sampling overhead at the device's error rates was restrictive for these circuit volumes, accepting a biased estimator at a much lower sampling cost.

  Methods

  The noise is not amplified by stretching pulses but by probabilistic error amplification against a learned noise model. A sparse Pauli-Lindblad model $e^{\mathcal{L}}$ is learned on the device for each of the three Pauli-twirled CNOT layers, giving local Pauli rates $\lambda_{l,i}$; multiplying those rates by $\alpha$ realises the overall noise channel at gain $G = \alpha + 1$, implemented by inserting products of one- and two-qubit Pauli errors before layer $l$ with probability $p_{l,i} = (1 - e^{-2(G-1)\lambda_{l,i}})/2$. Circuits were run at gains $G \in \{1, 1.2, 1.6\}$ (and $G = 1, 1.3, 1.6$ for panel a of Fig. 4), the per-gain estimates were fit by an extrapolating function in $G$ and evaluated at $G = 0$. Exponential extrapolation is preferred to linear where the per-gain estimates are well resolved, and the extrapolation model complexity is iteratively downgraded when expectation values sit unresolvably close to zero. Readout error mitigation is applied on top.

  Data

  The benchmark is the Trotterized time evolution of a 2D transverse-field Ising model sharing the topology of the processor, from the initial state $|0\rangle^{\otimes 127}$, with $\theta_J = -2J\delta t = -\pi/2$ chosen so each $ZZ$ rotation costs a single CNOT and the transverse-field angle $\theta_h$ swept between 0 and $\pi/2$; depths run up to 20 Trotter steps -- 60 layers of two-qubit gates, 2,880 CNOT gates in total. The noise model that supplies the amplification is learned on the device itself by characterization experiments rather than assumed. The Data availability statement gives the datasets generated and analysed as https://doi.org/10.6084/m9.figshare.22500355.

  Code

  None found yet.

  Results

  Primarily the IBM Eagle processor ibm_kyiv -- the paper's own word, and the only device it names: 127 fixed-frequency transmon qubits, heavy-hex connectivity, median $T_1$ and $T_2$ of 288 $\mu$s and 127 $\mu$s. At the Clifford point $\theta_h = 0$, where every $\langle Z_q\rangle$ is ideally 1, 2,000 circuit instances were generated for each $G$ and each executed 64 times, totalling 384,000 executions; the unmitigated magnetization decays monotonically with depth while ZNE tracks the ideal value, with a small bias, out to 20 Trotter steps or 60 CNOT depth. For non-Clifford circuits at five Trotter steps (15 CNOT layers), mitigated weight-1, weight-10 and weight-17 observables track the exact evolution across the $\theta_h$ sweep; the weight-17 observable's light cone expands to 68 qubits, past brute-force simulation, so the comparison there is against tensor networks. In the regime beyond exact verification the paper states both halves of its own run-time comparison: the tensor-network simulations of Fig. 4 ran on a 64-core, 2.45-GHz processor with 128 GB of memory, taking 8 h to reach one data point at fixed $\theta_h$ for Fig. 4a and 30 h for Fig. 4b, while the corresponding quantum wall-clock run time was approximately 4 h for Fig. 4a and 9.5 h for Fig. 4b, dominated by classical processing delays. The paper separately estimates the device run time for the mitigated expectation values at 614,400 samples and a conservative 2 kHz sampling rate as 5 min 7 s.

• Zero-noise-extrapolated VQE · Qiskit

  From the repository — run, not written up from a paper · unsupported

  About

  Energy estimates at amplified noise levels are extrapolated toward an inferred zero-noise limit.

  Methods

  None found yet.

  Data

  None found yet.

  Code

  Qiskit

  Results

  Literature-backed method record; algorithmic scope and evidence boundary reviewed, with no benchmark run claimed.

• Zero-noise-extrapolated VQE

  Energy estimates at amplified noise levels are extrapolated toward an inferred zero-noise limit.