Exponential Error Suppression for Near-Term Quantum Devices

Suppressing noise in physical systems is of fundamental importance. As quantum computers mature, quantum error correcting codes (QECs) will be adopted in order to suppress errors to any desired level. However in the noisy, intermediate-scale quantum (NISQ) era, the complexity and scale required to adopt even the smallest QEC is prohibitive: a single logical qubit needs to be encoded into many thousands of physical qubits. Here we show that, for the crucial case of estimating expectation values of observables (key to almost all NISQ algorithms) one can indeed achieve an effective exponential suppression. We take $n$ independently prepared circuit outputs to create a state whose symmetries prevent errors from contributing bias to the expected value. The approach is very well suited for current and near-term quantum devices as it is modular in the main computation and requires only a shallow circuit that bridges the $n$ copies immediately prior to measurement. Using no more than four circuit copies, we confirm error suppression below ${10}^{-6}$ for circuits consisting of several hundred noisy gates (2-qubit gate error 0.5%) in numerical simulations validating our approach.

Popular Summary

Existing quantum computers can already simulate complex quantum states that cannot be reasonably tackled classically. However, imperfections (or noise) in their quantum operations severely limit their practical applicability. To reach their full potential, quantum computers will need to implement quantum error-correcting codes, which can efficiently suppress errors but, unfortunately, are prohibitively expensive. A single, noise-protected logical qubit is encoded into hundreds or thousands of physical qubits, and highly nontrivial additional measures are performed to maintain its logical quantum state. Here, we introduce a means to gain much of the benefit of quantum error-correcting codes in a way that is compatible with near-term devices.

Our technique works in the specific (but pivotal) case of estimating expectation values: With near-term quantum hardware, one typically aims to avoid lengthy quantum computations and instead estimates expectation values after a shallow state-preparation stage. The core idea is that we prepare identical copies of the shallow computation and then apply a compact process called a derangement circuit, which effectively uses the copies to validate one another. This approach does require an increase in the number of qubits (a small integer multiple, at least two), but in return, it provides exponential error suppression, which other mitigation techniques cannot.

This work motivates a new architectural model of near-term quantum computing whereby multiple noisy quantum processors perform the same quantum computation in parallel, and the copies are then bridged immediately prior to measurement by a derangement circuit that requires long-range, weak entangling links between the processors.

Figure 1

A possible implementation of the ESD approach. Our derangement operator $D_n$ is a generalization of the swap operator and acts on $n$ (not necessarily identical) copies of the quantum state $\rho =\lambda |\psi ⟩⟨\psi |+(1-\lambda ){\rho }_{\mathrm{err}}$. In the above circuit $D_n$ permutes the $n$ registers and allows only permutation symmetric combinations, e.g., $|\psi {⟩}^{\otimes n}$, to contribute to the expectation-value measurement process. The probability of measuring the ancilla qubit in the 0 state enables us to approximate the expectation value $⟨\psi |\sigma |\psi ⟩$ and errors are suppressed exponentially in $n$. This derangement operator can be implemented as a shallow circuit using a linear number $N(n-1)$ of primitive, controlled, 2-qubit swap gates.

Figure 2

Simulation of a 12-qubit state with 372 noisy quantum gates. Errors in estimating the expectation value in the dominant eigenvector $⟨\psi |\sigma |\psi ⟩$ decay exponentially with the number of copies $n$ of the quantum state $\rho$. Dashed lines: our general upper bounds on the errors from result 1 and result 2 only require the knowledge of the dominant eigenvalue $\lambda$ and the largest error probability $p_{\max }$ from Eq. (1). Solid lines: our upper bounds based on the Rényi entropy of the quantum states. Red and blue bars: approximation errors obtained with 500 randomly selected Pauli strings (observables $\sigma$) are significantly below the upper bounds. (left) All copies of $\rho$ are perfectly identical and (right) all copies of $\rho$ are significantly different (but they commute as this case can be simulated efficiently) and their trace distance is 0.01 due to their different eigenvalue distributions.

Figure 3

Mitigating errors in the derangement operator. Extrapolation errors using various different fitting techniques versus the number of fitting points for 50 randomly selected Ansatz states. Elementary gates in the derangement operator in Fig. 1 have an error rate $\epsilon ={10}^{-3}$ and an experimentalist can increase this error in $k=2,3,4,\dots$ steps up to $\epsilon ={10}^{-2}$. The probability ${\mathrm{prob}}_0$ from Fig. 1 at $\epsilon =0$ is estimated by extrapolating to $\epsilon =0$. The derangement measurement is highly resilient to imperfections (see text) and ${\mathrm{prob}}_0(\epsilon )$ is almost linear in $\epsilon$. Increasing the degree of the fitting polynomial (blue circles) reduces the extrapolation error exponentially.

Figure 4

(a) Ansatz used to prepare the ground state of the Hamiltonian in Eq. (8) for $N=6$ qubits. (b) Type B (type C) recompilation of controlled-swap (including observable) gates from Table 1 requires 5 (4) applications of the hardware-native entangling gates. (c) Error in estimating the ground-state energy with and without mitigation as a function of the number of expected errors $\xi$ in the Ansatz circuit. We assume that the experimentalist can amplify the vast majority of the noise (94%) in the hardware-native gates, but not all of it, limiting extrapolation to a finite precision (orange diamonds). Although the derangement circuit is also degraded by noise, it can still drastically reduce errors both in combination with (black crosses) and without extrapolation (magenta dots). Dashed lines correspond to $\mathrm{Tr}[\mathcal{H}{\rho }^n]/\mathrm{Tr}[{\rho }^n]$ as obtained via noiseless derangement circuits. When increasing $n$, we approach in exponential order a nonzero error (dashed gray line) which is due to the coherent mismatch in the dominant eigenvector. The present demonstration on $2\times 6+1$ qubits should rather be viewed as a worst-case scenario since increasing the scale of the computation will favor the ESD approach.

Figure 5

Coherent mismatch $c$ in the dominant eigenvector $\sqrt{1-c}|\psi ⟩+\sqrt{c}|{\psi }_{\mathrm{err}}⟩$ of the density matrix $\rho$ in case of a purely incoherent noise model (single- and 2-qubit depolarizing). We simulated the same 12-qubit system with 372 noisy gates from Fig. 2 in which single-qubit gates undergo single-qubit depolarization with probability $0.1\epsilon$ while 2-qubit gates undergo 2-qubit depolarization with probability $\epsilon$. (a) The coherent mismatch is small and scales with the per-gate error $\epsilon$ as $c=\mathcal{O}({\epsilon }^2)$ as explained in the text. (b) The coherent mismatch is small and it grows with the number of gates $\nu$ at a fixed gate error $\epsilon ={10}^{-3}$ as $c=\mathcal{O}(\nu )$.

Figure 6

(a) The fidelity ${\eta }_1$ decreases purely due to the coherent mismatch while the fidelity ${\eta }_2$ decays purely due to the incoherent effect of the noise channel as discussed in Appendix pp3-s1. The ratio ${\eta }_2/{\eta }_1$ appears to decay exponentially in the investigated region (linear in the logarithmic plot) when we increase the number of gates for the fixed 2-qubit gate error $\epsilon ={10}^{-2}$ and this ensures us that the coherent mismatch in the dominant eigenvector becomes negligible for large systems (large $\nu$). We simulated the same circuit as in Fig. 5. (b) Mitigating the error caused by a coherent mismatch in the dominant eigenvector of the density matrix as discussed in Appendix pp3-s1. Gate error levels in the state-preparation stage were varied in $k=2,3,4,\dots$ steps and the obtained expectation values were extrapolated to the zero error limit. The red horizontal line represents the error bound from result 1, and one can straightforwardly suppress the effect of the coherent mismatch below this error bound by fitting low-order polynomials. The error saturates when increasing the degree of the fitting polynomial and can only be further reduced by increasing the number $n$ of copies.

Figure 7

Recompiling the controlled-swap gate into hardware-native gates. In all circuits the last $R_z$ rotation gate on the control qubit can be removed as it commutes with the controlled-swap gate and can be merged with the basis transformation of the ancilla qubit (Hadamard gate) immediately prior to measurement in Fig. 1.

Figure 8

Recompiling the controlled-swap gate assuming that the hardware can natively implement 3-qubit gates, such as the $xxx$ gate (first row) and the controlled-controlled-phase gate (second row).

Figure 9

Type C: recompiling the product of the controlled-swap gate and the controlled observable $\sigma$ up to a local SU(4) freedom on the swapped qubits.

Figure 10

Left: reducing errors in the derangement operator $D_2$ via generalized twirling operations as discussed in Appendix pp5-s2. Errors in measuring expectation values of 50 randomly selected observables $\sigma \in \{{\mathrm{Id}}_2,X,Y,Z{\}}^{\otimes N}$ were computed with ($\tilde{\mathcal{E}}$) and without ($\mathcal{E}$) twirling. The median of the ratios $F$ of the two errors $F≔\tilde{\mathcal{E}}/\mathcal{E}$ is plotted as a function of the number of expected errors in the derangement circuit ($\xi$)—shading represents the quantiles $1/4$ and $3/4$. Twirling reduces 30%–50% of errors in the noisy derangement circuit, but it can be viewed as a worst-case scenario as discussed in Appendix pp5-s2. Right: finding the ground-state energy $E$ of the spin-ring Hamiltonian from Eq. (8) using the variational Hamiltonian Ansatz with $l$ layers as discussed in Appendix pp6-s3. The Ansatz parameters were optimized in 5 independent instances and the average of the distance from the ground-state energy $\mathrm{\Delta }E$ is plotted as a function of $l$. Shading represents the minimum of the 5 instances.

Figure 11

Example of a 2-block Ansatz circuit of 8 qubits. We used a 10-block circuit of 12 qubits in our simulations in Fig. 2 consisting of overall 372 noisy gates. Each 2-qubit gate undergoes 2-qubit depolarizing noise with 0.5% probability and each single-qubit gate undergoes depolarizing noise with 0.05% probability.

Figure 12

Circuit that we simulate in Fig. 3. A controlled-swap gate acting on qubits $k$, $l$, and $m$ is followed by 2-qubit depolarizations between qubits $k$, $l$ and $k$, $m$ and $l$, $m$. A variant in which damping errors follow the depolarizations effectively resulted in the same error mitigation performance.