Quantum Error Mitigation

A noisy quantum processor gives you the wrong answer, but not randomly wrong: the error is a systematic bias whose structure you can learn. Quantum error mitigation is the discipline of measuring that bias and subtracting it from a classical number — almost always an expectation value $\langle O\rangle=\mathrm{Tr}(O\rho_{\mathrm{ideal}})$ — after the circuit has run. The hardware never realizes the ideal state $\rho_{\mathrm{ideal}}$ ; it realizes $\mathcal{N}(\rho_{\mathrm{ideal}})$ for some completely positive trace-preserving noise map $\mathcal{N}$ , and the measured average $\mathrm{Tr}(O\,\mathcal{N}(\rho_{\mathrm{ideal}}))$ is offset from the target. Mitigation uses a model of $\mathcal{N}$ to build an estimator that is unbiased even though every circuit you actually ran was noisy. The sharp line to draw at the outset is against quantum error correction. Error correction stores logical information redundantly across many physical qubits and actively restores the quantum state fault-tolerantly, with provably scalable thresholds — at the price of enormous qubit and gate overhead the near-term machines do not have. Mitigation restores no state and protects no coherence; it de-biases a single classical estimate. And it pays for that de-biasing in one hard currency: statistical variance. Every genuine cancellation of noise inflates the spread of the estimator, so the number of circuit repetitions must grow — exponentially in circuit volume once the product of error rate and depth stops being small. Two techniques anchor the field. Zero-noise extrapolation deliberately amplifies the noise by known factors and extrapolates the trend back to zero. Probabilistic error cancellation implements the unphysical inverse channel $\mathcal{N}^{-1}$ as a signed average over physically realizable circuits. Both, and every cousin — virtual distillation, symmetry verification, measurement-error mitigation — are the same trade written differently: bias bought with variance.

Key takeaways

Quantum error mitigation removes the bias in a noisy expectation value $\langle O\rangle$ by classical post-processing plus deliberate circuit modification, with none of the qubit overhead of fault-tolerant error correction.
It does not protect the quantum state, unlike error correction; it de-biases only the final classical estimate, trading systematic bias for sampling variance.
Zero-noise extrapolation (ZNE) scales the noise by factors $\lambda\ge 1$ (e.g. by gate folding) and extrapolates the measured $\langle O\rangle(\lambda)$ back to $\lambda\to 0$ ; two-point Richardson gives $a_0=\tfrac32\langle O\rangle(1)-\tfrac12\langle O\rangle(3)$ .
Probabilistic error cancellation (PEC) inverts a characterized channel via a quasiprobability decomposition $\mathcal{E}^{-1}=\sum_i\eta_i\mathcal{B}_i$ with $\sum_i\eta_i=1$ ; the cost is set by the 1-norm $\gamma=\sum_i|\eta_i|\ge 1$ .
The unavoidable price is variance: PEC scales the shot count as $\gamma^2$ , and depolarizing rescaling by fidelity $p$ scales it as $1/p^2$ , so QEM is a near-term, not a scalable, strategy.
Worked number: a per-gate $\gamma=1.0625$ at 3 percent depolarizing error compounds to $\gamma\approx430$ over 100 gates, a shot overhead $\gamma^2\approx1.8\times10^5$ .
Key idea: mitigation does not fix the state; it fixes the number. Using a known noise model, one builds an unbiased estimator of $\mathrm{Tr}(O\rho_{\mathrm{ideal}})$ as a signed combination of measurements on realizable noisy circuits — paying entirely in sampling variance, which scales as $\gamma^2$ in the channel's quasiprobability 1-norm.

The core insight

Mitigation does not fix the state; it fixes the number. By exploiting a known model of the noise channel, one builds an unbiased (or less biased) estimator of $\mathrm{Tr}(O\rho_{\mathrm{ideal}})$ as a signed/weighted combination of measurements on physically realizable noisy circuits — paying entirely in sampling variance, which scales as $\gamma^2$ in the channel's quasiprobability 1-norm.

Where this lives today

Mitigation moved from proposal to headline in 2023, when an IBM team led by Youngseok Kim ran a 127-qubit kicked-Ising dynamics circuit — beyond what brute-force classical simulation could then verify — and used zero-noise extrapolation with a learned Pauli-noise model (probabilistic error amplification) to extract magnetization observables that matched later tensor-network checks. Billed as "evidence for the utility of quantum computing before fault tolerance," it was the clearest demonstration that characterized noise plus disciplined extrapolation can pull a trustworthy number out of a circuit whose raw output is badly biased. The same toolbox is now standard in variational quantum eigensolvers, where each Pauli term of a Hamiltonian is mitigated before the energy is assembled, and in Trotterized dynamics, where observables are extrapolated to the noiseless trajectory while $\epsilon d\lesssim 1$ . Measurement-error mitigation — inverting a readout response matrix — ships in every major SDK as the cheapest member of the family. The live research questions are the boundaries this article draws: how to learn the noise model accurately enough that the reported error bar reflects the true bias, and how far the exponential sampling wall can be pushed back before fault-tolerant qubits make the enterprise unnecessary.

The physical picture

Read the noisy expectation value as a true signal sitting under a systematic, model-dependent offset. Zero-noise extrapolation attacks the offset by making it bigger in a controlled way. If you can dial the effective noise strength up to $\lambda=3$ — by stretching gate durations, or by replacing each gate $G$ with $G\,G^\dagger G$ so three noisy gates do the work of one — you get a curve $\langle O\rangle(\lambda)$ and read off where it points as $\lambda\to0$ . It is the same move a physicist makes extrapolating a measured quantity to zero temperature: you never reach the ideal point, but the trend through the points you can reach tells you where it lies. The one assumption is that the dependence on $\lambda$ is smooth and low-order — true for Markovian noise, and the place where the method quietly fails when it is not. Probabilistic error cancellation takes the opposite stance: what linear combination of realizable noisy circuits reproduces, on average, the output of the ideal one? Since the ideal channel is the noisy channel composed with its own inverse, and that inverse is not a physical (completely positive) map, the combination is forced to carry negative weights. You cannot run a circuit "a negative fraction of the time," but you can sample a circuit and flip the sign of its measured outcome before averaging. That single trick — sampling from $|\eta_i|/\gamma$ and multiplying by $\mathrm{sgn}(\eta_i)$ — makes PEC exactly unbiased and exactly expensive: the sign flips partially cancel, so the average converges slowly, at a rate measured by $\gamma=\sum_i|\eta_i|$ . The unifying picture is one line. Every mitigation method writes $\langle O\rangle_{\mathrm{ideal}}=\sum_k c_k\,\langle O\rangle^{(k)}_{\mathrm{noisy}}$ , a signed combination of measurable quantities. The bias vanishes when the $c_k$ come from a correct noise model; the variance grows with $\sum_k|c_k|$ or its square. Mitigation is a controlled exchange of accuracy for shots. The first figure shows the ZNE half of that trade with real numbers; the second shows what the variance side costs as the circuit grows.

Figure

Zero-noise extrapolation of a decaying observable: gate folding measures ⟨O⟩ = 0.655 at noise scale λ = 1 and 0.439 at λ = 3, a linear Richardson fit extrapolates to 0.763 while the true noiseless value is 0.800, so an exponential fit recovers 0.800 exactly — the model choice sets the residual bias

The second figure is the reason no one runs mitigation on a thousand-gate circuit. The variance overhead multiplies gate by gate: a per-gate 1-norm $\gamma$ only slightly above one, compounded over hundreds of gates, becomes an astronomical factor. This is the exponential wall, drawn on a logarithmic axis where the exponential-in-depth cost appears as a straight line — and it is exactly why quantum error mitigation buys the near term but cannot replace fault tolerance.

Figure

The sampling wall of probabilistic error cancellation: with per-gate 1-norm γ = 1.0625 at 3 percent depolarizing error, the shot overhead γ² grows exponentially with circuit size — about 10⁵ extra shots at 100 gates and 3 × 10¹⁰ at 200 gates, which is why mitigation is a near-term and not a scalable strategy

The core equations

The starting point is the gap the whole field exists to close: the number you want and the number the machine gives you. The target is a trace against the ideal state; the measurement is the same trace against the noisy state $\mathcal{N}(\rho_{\mathrm{ideal}})$ , and their difference is the bias mitigation must remove.

\langle O\rangle_{\mathrm{ideal}} = \mathrm{Tr}\!\big(O\,\rho_{\mathrm{ideal}}\big),\qquad \langle O\rangle_{\mathrm{noisy}} = \mathrm{Tr}\!\big(O\,\mathcal{N}(\rho_{\mathrm{ideal}})\big) \qquad (1)

where: $O$ — Hermitian observable whose expectation value is the quantity of interest; $\rho_{\mathrm{ideal}}$ — density operator produced by the noiseless circuit; $\mathcal{N}$ — CPTP map describing the aggregate noise of the circuit; $\mathrm{Tr}$ — matrix trace. Reading: mitigation never touches $\rho$ ; it manipulates the numbers on the right to reconstruct the number on the left. Zero-noise extrapolation models the measured value as a smooth function of a dialable noise strength $\lambda$ (with $\lambda=1$ the bare hardware) and reads off the intercept. Keeping only the linear term and measuring at two scales gives the two-point Richardson estimator, the workhorse of practical ZNE.

\langle O\rangle(\lambda) = a_0 + a_1\lambda + a_2\lambda^2 + \cdots,\qquad \hat{\langle O\rangle}_{\mathrm{lin}} = \frac{\lambda_2\,\langle O\rangle(\lambda_1) - \lambda_1\,\langle O\rangle(\lambda_2)}{\lambda_2 - \lambda_1} \qquad (2)

where: $\lambda$ — noise scale factor; $a_0=\langle O\rangle_{\mathrm{ideal}}$ — the extrapolation target; $a_k$ — higher fit coefficients; $\lambda_1,\lambda_2$ — two scales, typically $1$ and $3$ from gate folding. Reading: for $\lambda_1=1,\lambda_2=3$ this is $a_0=\tfrac32\langle O\rangle(1)-\tfrac12\langle O\rangle(3)$ , and the squared coefficients $(\tfrac32)^2+(\tfrac12)^2=2.5$ are the price in variance. Probabilistic error cancellation expands the formal inverse of the noisy gate in a basis of implementable operations. The coefficients are real but not all positive, they sum to one (trace preservation), and their absolute sum $\gamma$ is the per-gate sampling overhead. Sampling from the normalized magnitudes and attaching the signs yields an exactly unbiased estimator.

\mathcal{E}^{-1}=\sum_i \eta_i\,\mathcal{B}_i,\ \ \sum_i \eta_i=1,\ \ \gamma=\sum_i|\eta_i|\ge 1;\qquad \langle O\rangle_{\mathrm{ideal}} = \gamma\,\mathbb{E}_{i\sim p_i}\!\big[\mathrm{sgn}(\eta_i)\,O_i\big],\ \ p_i=\frac{|\eta_i|}{\gamma} \qquad (3)

where: $\mathcal{E}^{-1}$ — formal inverse of the noisy gate implementation; $\mathcal{B}_i$ — realizable basis operations (Paulis, easy CPTP maps); $\eta_i$ — real quasiprobability weights; $\gamma$ — 1-norm and overhead factor; $p_i$ — sampling distribution; $O_i$ — single-shot outcome with $\mathcal{B}_i$ inserted. Reading: negativity of some $\eta_i$ forces $\gamma>1$ , and $\gamma=1$ only if the gate is already noiseless. The cost is universal and quantitative. Because each sampled contribution has magnitude at most $\gamma\lVert O\rVert_\infty$ , the estimator's variance carries a factor $\gamma^2$ , so the shots needed for a target precision $\delta$ scale as $\gamma^2/\delta^2$ . For the special case of a global depolarizing channel of fidelity $p$ , the same logic reduces to dividing out $p$ , with variance blown up by $1/p^2$ .

\mathrm{Var}\big[\hat{\langle O\rangle}\big]\le \frac{\gamma^2\lVert O\rVert_\infty^2}{N},\quad N\sim\frac{\gamma^2}{\delta^2};\qquad \langle O\rangle_{\mathrm{ideal}}=\frac{\langle O\rangle_{\mathrm{noisy}}-(1-p)\,\mathrm{Tr}(O)/2^{n}}{p} \qquad (4)

where: $N$ — number of shots; $\delta$ — target additive precision; $\lVert O\rVert_\infty$ — spectral norm of $O$ ; $p$ — depolarizing fidelity, $n$ qubits so $2^n$ is the dimension. Reading: over a depth- $d$ circuit $\gamma\sim e^{2\epsilon d}$ and $p\sim e^{-\epsilon d}$ , so both routes give a shot count exponential in circuit volume — the wall of the second figure.

Derivation AExistence and 1-norm cost of probabilistic error cancellation

The goal is to prove that any invertible noisy gate can be corrected exactly by a purely classical post-processing recipe run on physically legal circuits, and to pin down what that exactness costs. We want a sampling procedure whose average reproduces the ideal channel, built only from operations the hardware can perform, that is unbiased for any target precision given an exact noise model and needs a number of shots scaling as $\gamma^2/\delta^2$ in the quasiprobability 1-norm $\gamma$ . The argument is four moves: insert the inverse, expand it in a realizable basis, repackage the signed weights as a probability distribution with signs, and bound the variance.

Step 1. Write the ideal gate as the noisy gate cleaned up by its own inverse, $\mathcal{U}_{\mathrm{ideal}}=\mathcal{E}^{-1}\circ\mathcal{E}$ , where $\mathcal{E}$ is the noisy implementation the hardware runs. Why: this is the only sense in which "undoing the noise" is exact — we are not modifying $\mathcal{E}$ , we are prepending a correction $\mathcal{E}^{-1}$ that lives entirely in post-processing.

Step 2. Expand the inverse in a fixed basis of implementable maps, $\mathcal{E}^{-1}=\sum_i\eta_i\,\mathcal{B}_i$ with $\eta_i\in\mathbb{R}$ . For Pauli noise the $\mathcal{B}_i$ are Pauli conjugations and the $\eta_i$ are read off by inverting the (diagonal) Pauli transfer matrix; trace preservation of $\mathcal{E}^{-1}$ forces $\sum_i\eta_i=1$ . Why: $\mathcal{E}^{-1}$ is a legitimate linear map but not a legitimate channel — it is not completely positive — so it cannot be run directly, yet it can always be written as a real combination of things that can.

Step 3. Split each weight into a magnitude and a sign: define $\gamma=\sum_i|\eta_i|$ , $p_i=|\eta_i|/\gamma$ , and $\sigma_i=\mathrm{sgn}(\eta_i)$ . The $p_i$ form a genuine probability distribution, and because some $\eta_i<0$ whenever the noise is nontrivial, $\gamma>1$ . Why: this is the sleight of hand that makes a signed sum runnable — you sample the circuit $\mathcal{B}_i$ with probability $p_i$ and remember to flip the outcome's sign when $\sigma_i=-1$ .

Step 4. Rewrite the target expectation, $\langle O\rangle_{\mathrm{ideal}}=\sum_i\eta_i\,\mathrm{Tr}(O\,\mathcal{B}_i\mathcal{E}\rho)=\gamma\sum_i p_i\,\sigma_i\,\langle O\rangle_i$ , and check the mean of the single-shot estimator $\gamma\sigma_i O_i$ : it is $\gamma\sum_i p_i\sigma_i\langle O\rangle_i=\langle O\rangle_{\mathrm{ideal}}$ . Why: the estimator is exactly unbiased — there is no truncation, no model order to converge — provided the noise model behind the $\eta_i$ is correct.

Step 5. Bound the spread. Since $|\gamma\sigma_i O_i|\le\gamma\lVert O\rVert_\infty$ , Hoeffding gives $\mathrm{Var}\le\gamma^2\lVert O\rVert_\infty^2$ and a shot count $N=O(\gamma^2/\delta^2)$ ; over a depth- $d$ circuit the per-gate factors multiply, $\gamma=\prod_g\gamma_g\sim e^{2\epsilon d}$ . Why: exactness is not free — the sign flips make contributions partially cancel, so the variance carries the full $\gamma^2$ , and that factor is exponential in circuit volume. *Note: $\gamma\ge1$ with equality if and only if $\mathcal{E}$ is already the target operation, and $\sum_i\eta_i=1$ enforces trace preservation.*

Go deeper · Derivation B · 1 minWhy mitigation must amplify variance — the bias–variance trade

Derivation BWhy mitigation must amplify variance — the bias–variance trade

The goal is to show that variance amplification is not an artifact of PEC but a theorem about any linear de-biasing scheme, and to compute the amplification exactly for the cleanest noise model. Take the dominant noise to be a depolarizing contraction that shrinks a traceless observable's expectation by a fidelity factor $p<1$ , so the noisy estimate is a faithful but attenuated copy of the ideal one. Recovering the ideal value means dividing the empirical mean by $p$ ; propagating the fluctuations through that division gives a hard floor — the variance is multiplied by $1/p^2$ , and since $p$ decays exponentially with circuit volume, so does the shot budget.

Step 1. Bound the raw single-shot spread: for a bounded observable $\mathrm{Var}[O_{\mathrm{noisy}}]\le\lVert O\rVert_\infty^2$ , and for a projective measurement it is at most $1$ . Why: this is the variance you would already pay with no mitigation at all — the baseline the de-biasing multiplies.

Step 2. Apply the depolarizing rescaling. For a traceless $O$ the noisy mean is $\langle O\rangle_{\mathrm{noisy}}=p\,\langle O\rangle_{\mathrm{ideal}}$ , so the unbiased estimator is $\hat{\langle O\rangle}_{\mathrm{mit}}=\overline{O_{\mathrm{noisy}}}/p$ . Why: recovering the ideal number literally requires multiplying the data by $1/p>1$ , and whatever you do to a random variable, you do to its fluctuations.

Step 3. Propagate the variance through the scaling: $\mathrm{Var}[\hat{\langle O\rangle}_{\mathrm{mit}}]=\tfrac{1}{p^2}\,\mathrm{Var}[\overline{O_{\mathrm{noisy}}}]=\tfrac{1}{p^2}\,\mathrm{Var}[O_{\mathrm{noisy}}]/N$ . Why: scaling a variable by $1/p$ scales its variance by $1/p^2$ — this is the unavoidable, model-independent cost of de-biasing by a contraction factor.

Step 4. Read off the shot overhead: to hold the mitigated variance at the level of an ideal machine you need $N_{\mathrm{mit}}=N_{\mathrm{ideal}}/p^2$ , and with $p\sim e^{-\epsilon d}$ over a depth- $d$ circuit this is $N_{\mathrm{mit}}\sim e^{2\epsilon d}N_{\mathrm{ideal}}$ . Why: the fidelity of a whole circuit decays exponentially in its volume, so the $1/p^2$ overhead is exponential too — the wall that prevents QEM from scaling. *Note: the $1/p^2$ factor derived here is the depolarizing special case of the general $\gamma^2$ overhead.*

Go deeper · Derivation C · 1 minWhy a smooth expansion in noise strength exists

Derivation CWhy a smooth expansion in noise strength exists

The goal is to justify the central premise of zero-noise extrapolation — that $\langle O\rangle$ is an analytic, low-order function of the noise scale $\lambda$ — rather than assuming it. If the noise is Markovian it is generated by a Lindblad superoperator $\mathcal{L}$ acting for an effective time, and scaling the noise means scaling that time; the expectation value then inherits a genuine power series in $\lambda$ whose zeroth-order coefficient is exactly the noiseless target. This licenses fitting a polynomial and taking the intercept — and shows where the licence expires, since the series is only useful while the higher terms are small, i.e. while $\epsilon d\lesssim 1$ .

Step 1. Model the scaled-noise state as dissipative evolution for a stretched time, $\rho(\lambda)=e^{\lambda\mathcal{L}t}\,\rho_{\mathrm{ideal}}$ , where $\mathcal{L}$ is the GKSL generator and $\lambda$ multiplies the noise. Why: gate folding and pulse stretching physically lengthen the exposure to dissipation, which is exactly multiplying the generator's action by $\lambda$ .

Step 2. Expand the exponential, $\rho(\lambda)=\rho_{\mathrm{ideal}}+\lambda t\,\mathcal{L}\rho_{\mathrm{ideal}}+\tfrac12\lambda^2 t^2\,\mathcal{L}^2\rho_{\mathrm{ideal}}+\cdots$ . Why: the exponential is analytic in $\lambda$ , so the series converges and its truncation order is the only source of residual model bias.

Step 3. Take the trace with $O$ : $\langle O\rangle(\lambda)=a_0+a_1\lambda+a_2\lambda^2+\cdots$ with $a_k=\tfrac{t^k}{k!}\mathrm{Tr}(O\,\mathcal{L}^k\rho_{\mathrm{ideal}})$ , and in particular $a_0=\mathrm{Tr}(O\rho_{\mathrm{ideal}})$ . Why: linearity of the trace carries the state's series straight onto the observable, and the intercept $a_0$ is provably the noiseless value — the thing ZNE aims to recover. *Note: $a_0=\langle O\rangle_{\mathrm{ideal}}$ by construction; the polynomial premise holds for Markovian noise and degrades for non-Markovian or strongly coherent noise, where $\mathcal{L}$ is not a fixed generator.*

Limiting cases

Mitigation behaves so differently across regimes that quoting a single "overhead" is meaningless; the controlling parameter is the product of per-gate error rate $\epsilon$ and circuit depth $d$ .

Case	Result	Meaning
Weak noise, low depth ( $\epsilon d\ll 1$ )	$\gamma\approx 1+O(\epsilon d)$	Quasiprobability 1-norm and extrapolation factors are near unity; mitigation is cheap and nearly unbiased. The regime where QEM is genuinely useful.
Global depolarizing noise	$\langle O\rangle_{\mathrm{noisy}}=p\,\langle O\rangle_{\mathrm{ideal}}+(1-p)\mathrm{Tr}(O)/2^{n}$	A single fidelity $p$ suffices; mitigation reduces to dividing out $p$ , with overhead $1/p^2$ . Tractable and the basis of many benchmarks.
Large circuit volume ( $\epsilon d\gtrsim 1$ )	$\gamma\sim e^{2\epsilon d},\ N\sim\gamma^2\sim e^{4\epsilon d}$	Overhead is exponential, and the signal itself decays to the trace term, so the estimator divides a tiny number by a tiny number. QEM provably fails to scale here.
Coherent / non-Markovian noise	$\mathcal{N}_t\ne\mathcal{N}_{t_1}\circ\mathcal{N}_{t_2}$ (no semigroup)	Noise does not compose simply, so a single-cycle model amplified or inverted is biased; extrapolation forms and PEC inverses must be recharacterized and residual model bias dominates.

Common trapQEM is error correction on a budget

It is not. Quantum error mitigation does not improve the fidelity of any quantum state, does not extend the length of a coherent computation, and has no threshold theorem behind it. It de-biases a classical estimate and does so by burning shots — variance overhead that grows, generically exponentially, with circuit volume, so it cannot underwrite large fault-tolerant algorithms. Two further traps hide inside a working pipeline. First, mitigation is only as unbiased as the noise model it assumes: a mischaracterized channel produces a confidently wrong answer, often with a deceptively tight error bar, because the variance you report knows nothing about the model bias you missed. Second, extrapolation silently assumes a functional form for $\langle O\rangle(\lambda)$ — linear, exponential, Richardson — and the wrong form leaves a residual bias that survives even infinite statistics, as the 0.763-versus-0.800 gap in the first figure shows.

Worked examples

1 · Zero-noise extrapolation of a decaying observable (fully worked)

A circuit's target is $\langle O\rangle_{\mathrm{ideal}}=0.800$ . Under global depolarizing noise the true dependence on the noise scale is $\langle O\rangle(\lambda)=0.800\,e^{-0.2\lambda}$ , but the experimenter does not know this and measures only two points by gate folding: $\lambda=1$ and $\lambda=3$ . Compare a linear (Richardson) extrapolation with an exponential one, and quote the residual bias and variance cost of each.

Step 1. Take the two measurements. $\langle O\rangle(1)=0.800\,e^{-0.2}=0.800\times0.8187=\mathbf{0.655}$ and $\langle O\rangle(3)=0.800\,e^{-0.6}=0.800\times0.5488=\mathbf{0.439}$ . The bare hardware value $0.655$ is biased low by $0.145$ , an 18 percent error — that is what mitigation must fix.

Step 2. Linear Richardson with $\lambda_1=1,\lambda_2=3$ : $\hat{\langle O\rangle}_{\mathrm{lin}}=\tfrac32\langle O\rangle(1)-\tfrac12\langle O\rangle(3)=1.5(0.655)-0.5(0.439)=0.9825-0.2195=\mathbf{0.763}$ . The bias has fallen from $0.145$ to $0.037$ — a factor of four improvement — but it has not vanished, because a straight line cannot capture the true exponential curvature. The variance is amplified by $(\tfrac32)^2+(\tfrac12)^2=\mathbf{2.5}$ relative to a single measurement.

Step 3. Exponential fit $\langle O\rangle(\lambda)=A\,e^{-B\lambda}$ through the same two points. The ratio $0.439/0.655=0.670=e^{-2B}$ gives $B=-\tfrac12\ln 0.670=\mathbf{0.200}$ , and then $A=0.655\,e^{0.200}=0.655\times1.2214=\mathbf{0.800}$ . The exponential extrapolation recovers the true value exactly — because here it happens to be the correct functional form.

Step 4. Read the lesson. With identical data, linear ZNE leaves a $4.6$ percent residual bias while exponential ZNE leaves none; the difference is entirely the model, not the statistics. In practice one does not know the true form, so a real pipeline measures three or more scales and tests candidate fits — trading still more variance for confidence that the residual model bias is small. Linear → 0.763 (bias 0.037), exponential → 0.800 (bias 0), and the model choice, not the data, sets the accuracy.

2 · The quasiprobability overhead of a single depolarizing gate (fully worked)

A one-qubit gate suffers depolarizing noise $\Lambda(\rho)=(1-p)\rho+\tfrac{p}{3}(X\rho X+Y\rho Y+Z\rho Z)$ with $p=0.03$ . Build the PEC quasiprobability decomposition of $\Lambda^{-1}$ , compute the per-gate $\gamma$ , and find the shot overhead for a 100-gate circuit.

Step 1. Find how $\Lambda$ acts in the Pauli picture. Each non-identity Pauli component of the Bloch vector is scaled by $\lambda_{\mathrm{PTM}}=1-\tfrac{4p}{3}=1-0.04=\mathbf{0.96}$ (the identity component is untouched). So $\Lambda$ shrinks the Bloch vector by $0.96$ .

Step 2. Invert it. $\Lambda^{-1}$ must stretch the Bloch vector by $1/0.96=1.0417$ , which is a depolarizing-form map with a negative parameter $q$ : solving $1-\tfrac{4q}{3}=1/0.96$ gives $q=-p/(1-\tfrac{4p}{3})=-0.03/0.96=\mathbf{-0.03125}$ . The quasiprobability weights are $\eta_I=1-q=\mathbf{1.03125}$ and $\eta_X=\eta_Y=\eta_Z=q/3=\mathbf{-0.01042}$ .

Step 3. Check trace preservation and compute the 1-norm. The weights sum to $1.03125+3(-0.01042)=1.000$ ✓, and $\gamma=|1.03125|+3|0.01042|=1.03125+0.03125=\mathbf{1.0625}$ . The negativity of the three Pauli weights — implemented by sign-flipping those shots — is exactly what pushes $\gamma$ above one.

Step 4. Compound over the circuit. A depth-100 circuit has $\gamma_{\mathrm{tot}}=1.0625^{100}\approx\mathbf{430}$ , so the shot overhead is $\gamma_{\mathrm{tot}}^2\approx\mathbf{1.8\times10^{5}}$ — reaching the precision of the unmitigated machine now costs about $180{,}000$ times as many shots. (Cross-check: $\gamma_{\mathrm{gate}}\approx1+2p=e^{2p}$ per gate gives $e^{2\times0.03\times100}=e^{6}\approx403$ , matching the exact $430$ .) Per-gate γ = 1.0625, circuit γ ≈ 430, shot overhead ≈ 1.8 × 10⁵ — the exponential wall in a single number.

3 · Global depolarizing rescaling (quick)

An $n$ -qubit circuit is dominated by global depolarizing noise of fidelity $p=0.60$ . A traceless observable reads $\langle O\rangle_{\mathrm{noisy}}=0.48$ . Mitigate it and quote the shot cost.

Step 1. For traceless $O$ the background term $(1-p)\mathrm{Tr}(O)/2^n$ vanishes, so equation (4) collapses to $\langle O\rangle_{\mathrm{ideal}}=\langle O\rangle_{\mathrm{noisy}}/p=0.48/0.60=\mathbf{0.80}$ .

Step 2. The variance is amplified by $1/p^2=1/0.36=\mathbf{2.78}$ , so recovering the ideal precision needs about $2.78\times$ the shots. Mitigated value 0.80, at 2.78× the shot budget — cheap here because $p$ is close to one, ruinous once $p\to0$ .

Physics of the portrait

Quantum error mitigation is unusual among the topics in this chapter in having no eighteenth- or nineteenth-century ancestor: it is a child of the noisy-intermediate-scale era, invented when it became clear that useful quantum hardware would arrive years before fault tolerance did.

Origins · 1 minquantum error mitigation and how this was discovered

quantum error mitigation

de-biasing ⟨O⟩ = Tr(Oρ) without full error correction, born 2017 · near-term NISQ quantum computing

The idea crystallized in 2017 in two independent papers. Kristan Temme, Sergey Bravyi, and Jay Gambetta, at IBM, published Error Mitigation for Short-Depth Quantum Circuits, introducing both pillars at once: Richardson zero-noise extrapolation and the quasiprobability method that became probabilistic error cancellation, with the 1-norm $\gamma$ as its exact sampling cost. The same year Ying Li and Simon Benjamin, at Oxford, proposed "error extrapolation" for variational simulators, reaching zero-noise extrapolation from the angle of variational eigensolvers. The two founding pictures — extrapolate the noise away, or invert it by signed sampling — have organized the field ever since. In 2018 Suguru Endo, Benjamin, and Li made PEC practical by tying the quasiprobability weights to gate-set tomography, so the $\eta_i$ could be measured rather than assumed; in 2019 Abhinav Kandala and collaborators used ZNE to extend the reach of an IBM processor on a chemistry problem, the first clear hardware win. A different lineage opened in 2021 when Bálint Koczor, and independently William Huggins and coworkers, introduced virtual distillation, measuring with respect to $\rho^M/\mathrm{Tr}(\rho^M)$ to suppress incoherent error exponentially in $M$ . What unites the program is the founders' insight, now a theorem: you cannot fix the state for free, but you can fix the number, and the bill always arrives as variance.

History

2017

— Temme, Bravyi and Gambetta introduce zero-noise extrapolation and probabilistic error cancellation

2017

— Li and Benjamin propose error extrapolation for variational quantum simulation

2018

— Endo, Benjamin and Li make PEC practical by tying quasiprobabilities to gate-set tomography

2019

— Kandala and coworkers demonstrate zero-noise extrapolation on IBM hardware for a chemistry problem

2021

— Koczor and Huggins and coworkers introduce virtual distillation for exponential error suppression

Going further

Texts. Nielsen and Chuang, Quantum Computation and Quantum Information, Ch. 8 (Quantum Operations) and Ch. 10 (Quantum Error Correction) · Preskill, Quantum Computation Lecture Notes, Ch. 3 (Foundations II: Measurement and Evolution) · Wilde, Quantum Information Theory, 2nd ed. (channels, diamond norm) · Watrous, The Theory of Quantum Information (completely positive maps and norms) · Kitaev, Shen and Vyalyi, Classical and Quantum Computation (quantum operations) · Mermin, Quantum Computer Science. The channel and trace identities (equations 1 and 4) follow from CPTP-map linearity; the depolarizing rescaling and $1/p^2$ variance factor are direct consequences. The PEC derivation rests on unbiasedness via $\sum_i\eta_i=1$ and a Hoeffding bound with range $\gamma\lVert O\rVert_\infty$ , giving $\gamma\ge1$ with equality iff the gate is noiseless. The single-gate depolarizing example yields $\lambda_{\mathrm{PTM}}=1-\tfrac{4p}{3}=0.96$ , $q=-0.03125$ , weights $(1.03125,-0.01042,-0.01042,-0.01042)$ summing to $1$ , $\gamma=1.0625$ , and $1.0625^{100}\approx430$ with overhead $\approx1.8\times10^5$ . The ZNE example gives $0.8e^{-0.2}=0.655$ , $0.8e^{-0.6}=0.439$ , linear Richardson $1.5(0.655)-0.5(0.439)=0.763$ , exponential fit $B=0.200$ , $A=0.800$ . No device-specific constants are quoted; all overhead scalings ( $\gamma^2$ , $e^{2\epsilon d}$ ) are model-level consequences.

Prerequisites. Completely Positive Trace-Preserving Maps and Quantum Operations · Kraus Representation of Quantum Channels · Depolarizing Channel · Pauli and Bit/Phase-Flip Channels · Lindblad / GKSL Master Equation. Leads to. Markovian vs Non-Markovian Dynamics · Decoherence · Relaxation and Dephasing: T1 and T2 Times.

Ask the tutor: Ask me to walk through the PEC quasiprobability weights for a specific noise channel, or to show exactly where the exponential sampling wall would stop your circuit.

Is quantum error mitigation just a cheaper quantum error correction?

No — they solve different problems and one cannot substitute for the other. Error correction protects the quantum state: it encodes logical information across many physical qubits and actively restores it during the computation, with a threshold theorem guaranteeing scalability once the physical error rate is below a critical value. Mitigation touches no state and has no threshold; it takes the biased classical output of a fully noisy circuit and de-biases that single number in post-processing, paying in extra shots. Mitigation is what you do before you have enough qubits for correction, and its cost grows exponentially with circuit volume, so it buys the near term rather than replacing fault tolerance.

How is the noise "amplified" in zero-noise extrapolation?

Two common knobs. Pulse stretching lengthens each gate's control pulse so the qubit spends longer exposed to its environment, multiplying the dissipative action by a chosen factor $\lambda$ . Gate folding does it digitally: replace a gate $G$ by $G\,G^\dagger G$ , which equals $G$ ideally but subjects the qubit to three noisy operations instead of one, tripling the error to $\lambda=3$ ; folding subsets of gates gives intermediate factors. Both rest on the premise that scaling the noise this way faithfully scales the underlying generator — which holds for Markovian noise and degrades when the noise is coherent or non-Markovian, since folding no longer reproduces a cleanly stretched Lindblad evolution.

Where does the number γ come from, and why is it always at least one?

$\gamma$ is the 1-norm of the quasiprobability decomposition, $\gamma=\sum_i|\eta_i|$ , where the $\eta_i$ are the real weights expressing the unphysical inverse channel $\mathcal{E}^{-1}$ as a combination of physically implementable operations $\mathcal{B}_i$ . Trace preservation forces $\sum_i\eta_i=1$ , so if every weight were non-negative you would have a genuine probability distribution and $\gamma=1$ . But the inverse of a noisy channel is not completely positive, which forces at least one $\eta_i$ negative whenever the noise is nontrivial; the negative parts add in absolute value, pushing $\gamma$ above one. That excess is precisely the sampling overhead: the estimator's variance carries a factor $\gamma^2$ , and $\gamma=1$ only for a noiseless gate.

Can mitigation give the wrong answer even with infinite shots?

Yes, and this is its subtlest danger. Infinite shots drive the statistical error to zero, but they do nothing about model error. Extrapolation assumes a functional form for $\langle O\rangle(\lambda)$ ; if the true dependence is exponential and you fit a line, you converge — with a beautifully tight error bar — to a biased value, exactly the 0.763-versus-0.800 gap in the worked example. Probabilistic error cancellation assumes the characterized channel equals the real one; if the tomography is off, the $\eta_i$ are wrong and the "unbiased" estimator is biased. Because the reported variance knows nothing about these mismatches, a mitigated result can be confidently and invisibly wrong. This is why noise characterization, not the mitigation formula, is where the real work lies.

Why do so many analyses use the global depolarizing model?

Because it collapses the whole problem to one number and is often a good effective description of a deep, scrambling circuit. A global depolarizing channel replaces the state with the maximally mixed state with probability $1-p$ , so any observable's expectation simply attenuates: $\langle O\rangle_{\mathrm{noisy}}=p\,\langle O\rangle_{\mathrm{ideal}}+(1-p)\mathrm{Tr}(O)/2^n$ , and for a traceless observable it is just $p\,\langle O\rangle_{\mathrm{ideal}}$ . Mitigation then reduces to dividing out $p$ , with overhead $1/p^2$ . Randomized compiling (Pauli twirling) deliberately tailors hardware noise toward this Pauli/depolarizing form precisely so that this clean rescaling applies, which is why the model is both a pedagogical staple and a practical target rather than a mere caricature.

What is virtual distillation doing differently from ZNE and PEC?

It exploits the structure of the noisy state rather than a model of the noise channel. Many error processes push $\rho$ toward the maximally mixed state, but its dominant eigenvector often stays close to the ideal pure state. Preparing $M$ identical copies and measuring $\mathrm{Tr}(O\rho^M)/\mathrm{Tr}(\rho^M)$ — via a controlled-derangement circuit that cyclically permutes the copies — projects onto that dominant component, suppressing incoherent error exponentially in $M$ . Unlike ZNE and PEC it needs no extrapolation form and no channel tomography, but it needs $M$ times the qubits and cannot remove coherent error that has rotated the dominant eigenvector itself. It is complementary: pair it with ZNE or PEC, which handle the coherent and systematic bias it leaves behind.