Enhancing quantum computer performance via symmetrization

We consider a set of $n$-qubit computations $\mathfrak{U}_{n}$, each including state initialization, some operator from $SU(2^{n})$, and final measurements. Let $\mathfrak{R}_{n}$ be a set of realizations (of all quantum computations in $\mathfrak{U}_{n}$) that represent gate-level quantum circuits with qubit assignment, initialization, measurements, postprocessing, and possibly implementation details such as pulse sequences specified. We define the function $\pi:\mathfrak{R}_{n}\to\mathfrak{U}_{n}$ that finds the computation $u$ performed by a given concrete realization $r$. We define the general symmetries of $\mathfrak{U}_{n}$, denoted $\Gamma(\mathfrak{U}_{n})$, as the set of functions $\gamma:\mathfrak{R}_{n}\to\mathfrak{R}_{n}$ that satisfy

That is, $\gamma\in\Gamma(\mathfrak{U}_{n})$ if and only if for all $r\in\mathfrak{R}_{n}$, whenever $\pi(r)=u$, $\pi(\gamma(r))=u$. In other words, applying $\gamma$ to any realization will produce the same quantum computation. We also define computation-specific symmetries, $\Gamma(\mathfrak{U}_{n},r)$, as the set of functions $\gamma:\mathfrak{R}_{n}\to\mathfrak{R}_{n}$ that satisfy $\pi(\gamma(r))=\pi(r)$ for a particular $r$. For example, general symmetries could be conjugations (in group-action sense) of gate-level circuits by qubit permutations. Namely, the initial state is replaced by its permutation, the gates are applied on permuted qubits, and the measurement results are permuted back. Examples of computation-specific symmetries are gate decompositions, permutations of commuting gates, the addition of gates that preserve a given state (e.g., before measurement), and changes of gates and measurements compensated by changes in postprocessing. When $\mathfrak{R}_{n}$ specifies pulse sequences, symmetries can replace them with physically equivalent ones.

By distributing the computation over multiple symmetries, we cancel out the effect of control inaccuracies without amplification of random errors. The steps of the procedure, as shown in Fig. 1, are then:

1. 1.

   Define symmetries $\Gamma$ and sample $\Gamma^{\prime}\subset\Gamma$.

2. 2.

   Generate circuit variants for $\Gamma^{\prime}$.

3. 3.

   Execute each variant on the QC hardware.

4. 4.

   Aggregate all measurement statistics.

We now consider why symmetrization works. Let inaccurate realizations $\tilde{r}_{u}$ be determined by instantaneous parameters of the physical system, such that $\pi(\tilde{r}_{u})=u+\delta u$. A key example is unitary under- or over-rotations of particular gates Maksymov et al. (2021a).

To mitigate the impact of inaccuracies, we consider $\gamma(\tilde{r}_{u})$ for multiple $\gamma\in\Gamma=\Gamma(\mathfrak{U}_{n},r_{u})$ so as to symmetrize the error term in $\pi(\gamma(\tilde{r}_{u}))=u+\delta u_{\Gamma\text{-inv}}+\delta u_{\gamma}$. In the absence of errors, all realizations $r_{u}$ of $u$ implement the same computation. In the presence of errors, we rely on symmetrization over multiple $\gamma\in\Gamma$ to produce a computation $u+\delta u_{\Gamma\text{-inv}}+\left<\delta u_{\gamma}\right>_{\Gamma}$. As long as we select an uncorrelated set of $\gamma$, $||\left<\delta u_{\gamma}\right>_{\Gamma}||\ll\left<||\delta u_{\gamma}||\right>_{\Gamma}$, and the cumulative effect of non-$\Gamma$-invariant errors is much reduced.

In practice, rather than aggregating all $\pi(\gamma(\tilde{r}_{u}))$, we consider the output states produced by $\pi(\gamma(\tilde{r}_{u}))$ and aggregate their measurement statistics (because, e.g., coherently adding two quantum states would require additional qubits). The impact of inaccuracies on an ideal distribution $\mathbf{h_{u}}\in\mathbb{R}^{+}(2^{n})$ may be expressed as $\mathbf{h_{u}}+\delta h_{\Gamma\text{-inv}}+\delta h_{\gamma}$. Aggregating measurement statistics can “enhance the contrast” between the target output states and erroneously observed states. The error terms may cancel out, but more typically they would be uncorrelated. For example, if $\mathbf{h_{u}}=(0,\dots,1_{k},\dots,0_{2^{n}})$ and $\Gamma=S_{2^{n}}$, then the symmetrized result would be

The term $\delta h_{\Gamma\text{-inv}}$ in Eq. 2 captures the remaining fully-depolarizing error channel, i.e., the effect of incoherent errors Takagi et al. (2022). This residual error can be reduced with aggregation techniques such as plurality voting, e.g., for $\mathbf{h_{u}}$ with $l$ output states of frequency $\frac{1}{l}$ if $\varepsilon<1-l/2^{n}$ as proven in Supplemental Materials.

As a concrete example, we demonstrate the effect of symmetrization on 4-qubit circuits with six two-qubit gates on different qubit pairs and random single-qubit gates mapped to eight ions (see Methods). We model gate miscalibrations as random under-rotations of multiple two-qubit gates fixed per qubit pair. We assume an average under-rotation across all qubit pairs causing a similar error for all variants (Fig. 2a). Symmetries $\Gamma$ are represented by eight random qubit assignments $\gamma$. For each qubit assignment, we simulated corresponding inaccurate realizations to obtain vectors $\mathbf{h_{u}}+\delta h_{\Gamma\text{-inv}}+\delta h_{\gamma}$. In Fig. 2, we illustrate 256-dimensional vectors for ideal, individual, and symmetrized results by plotting their two largest principal components (principal component analysis (PCA) was initially performed on $\{\delta h_{\Gamma\text{-inv}}+\delta h_{\gamma}\}$ vectors). In the first case, since all gates are under-rotated by some amount on average, the variants fail to symmetrize the errors because they are only exploring qubit assignment symmetries. Hence, error effects $\delta h_{\Gamma\text{-inv}}$ remain after aggregation as shown in Fig. 2a. In the second example (Fig. 2b), we use additional symmetries of gate decompositions, to zero out $\delta h_{\Gamma\text{-inv}}$. The effect of under-rotation in fully-entangling XX gates is addressed using an alternative implementation that combines phase-flipped XX^-1 gates with pairs of X-gates thus implementing the same ideal unitary but reversing the effect of under-rotation.

An aggregation strategy for measurement statistics is a procedure that combines measurement statics from multiple implementations of the same computation. Without errors, all implementations should produce identical statistics in the limit (with infinite repetition count). An aggregation strategy is considered stable for a given type of statistics if, provided a set of identical statistics of this type, it produces another copy. Aggregation by componentwise averaging is trivially stable for statistics of any type. Yet aggregation by voting is not. This can be seen for the probability distribution $(1-\varepsilon,\varepsilon)$ which voting-based aggregation brings closer to $(1,0)$ for $\varepsilon<1/2$. What makes aggregation strategies useful is that ($i$) they coerce arbitrary statistics to statistics of the desired type, ($ii$) they distill original statistics from multiple erroneous variants of the original. To this end, output probability distributions are analytically characterized for many quantum algorithms including Shor’s and Grover’s. The choice of aggregation is determined by the type of output probability distribution of a given quantum algorithm.

For best performance, we recommend aggregation by plurality voting for quantum algorithms with ideal measurement statistics comprising of $l$ outputs with frequencies $\frac{1}{l}$. Such algorithms have zero-frequency outputs and a subset of target outputs that needs to be determined. For algorithms with different measurement statistics, aggregation by averaging can be used to avoid distortion.

We evaluate the impact of symmetrization and the choice of aggregation strategy experimentally by comparing the results of unsymmetrized runs to symmetrized runs with componentwise averaging and plurality voting. We use the IonQ Aria trapped-ion quantum computer for these experiments, configured to utilize 20 addressable qubits. See methods for experimental details.

Performance is measured by Hellinger fidelity, defined as a statistical overlap $F_{H}=\left(\sum_{i}\sqrt{p_{i}q_{i}}\right)^{2}$ between the actual output statistics $p_{i}$ and the ideal result $q_{i}$ is computed via an error-free simulator. $F_{H}$ ranges from $0$ to $1$, with 0 capturing probability distributions that do not overlap, and 1 corresponding to a pair of identical distributions. Also known as the Bhattacharyya coefficient Bhattacharyya (1943), $F_{H}$ is commonly used to measure the discrepancy between probability distributions and is consistent with the definition of fidelity for quantum states.

We demonstrate the impact of symmetrization on a 13-qubit single-output QFT-based adder circuit Lubinski et al. (2021). In Fig. 3a we compare the largest output probabilities out of $2^{13}$ between the unsymmetrized histogram, symmetrized with componentwise averaging, and symmetrized with plurality voting. The first and largest value corresponds to the bitstring with ideal probability 1 while the rest should have otuput probability 0. Fig. 3b shows the change in the error bars with the number of shots. We observe that symmetrization with componentwise averaging does not improve the probability of the target bitstring but does reduce next-largest probabilities, which allows for a dramatic increase in the probability of the target state after plurality voting (from 1.5% to 95%). For a 15-qubit QFT-based adder, the boost exceeds 100$\times$.

Next, we examine the performance of symmetrization for several use cases shown in Fig. 4. All jobs had 2500 shots taken with output probability distributions that vary in the number of correct output states with nonzero probability, and thus benefit differently from different aggregation strategies. In Fig. 4a, we evaluate results for QFT-based adders, phase estimation, and amplitude estimation with a single output state Lubinski et al. (2021). We see that symmetrization with plurality voting significantly increases $F_{H}$ while symmetrized runs with componentwise averaging show no improvement. In Fig. 4b, we compare results for amplitude estimation and Monte Carlo sampling circuits before tracing out the ancillary qubits. Symmetrization with plurality voting still shows the strongest improvement in $F_{H}$ but componentwise averaging is also better than no symmetrization because it evens out the errors across the four target states. In Fig. 4c, we evaluate symmetrization on variational quantum eigensolver (VQE) and quantum machine learning (QML) circuits Zhu et al. (2022b). Those circuits have broader, irregular output distribution, so that symmetrization with componentwise averaging shows the best improvement while plurality voting can skew the results. Circuits with more peaked output probability distributions often benefit more from aggregation with plurality voting (see Methods).

Since using all possible symmetries for a given quantum computation is impractical, we need to sample from those symmetries. For an error-free quantum computation, it suffices to use the identity symmetry alone. Assuming a single inaccuracy of a known type, very few symmetries would be sufficient, regardless of the magnitude of inaccuracies or the number of qubits. As the dimensionality of the error space grows, more symmetries must be sampled.

We sample symmetries $\gamma$ to minimize $\left<\delta u_{\gamma}\right>$. Selecting dissimilar (rather than random Takagi et al. (2022)) symmetries reduces the bias and decorrelates inaccuracies between the variants. If symmetries $\Gamma$ are qubit assignments, one may select assignments that share fewer gates between physical qubits for a given device-specific connectivity.

Continuing the discussion in Section II.B, we compare two aggregation strategies for measurement statistics: one represents them by frequency distributions, and the other — by raw output samples.

Due to the nonlinearity of voting, it is a stable aggregation strategy for ideal output probabilities with $l$ equally probable outputs and $r-l$ zero frequencies (see Supplemental Materials for proofs). Relevant circuits include QFT-based adders, phase and amplitude estimation algorithms, some Monte Carlo algorithms.

We use the IonQ Aria IonQ (2022a) trapped-ion quantum computer which utilizes trapped Ytterbium ions individually addressed by pulses of 355 nm light. These pulses can be engineered to generate a Mølmer-Sørensen entangling gate between ions as well as single qubit rotations/gates. The Aria system uses 22 such ions as qubits to perform quantum information processing. Here, we split our experiments into 25 different maps (variants) between physical and computational qubits, running 100 experimental shots on each variant resulting in 2500 total experimental repetitions. For circuits on more than six qubits, we genrated permutations on a set of physical qubits. Otherwise, two additional physical qubits were utilized to increase the number of diverse mappings. All variants were measured under similar conditions. To this end, for most of our experiments, we executed our jobs within one calibration cycle. Whenever this was not possible (e.g. due to ion-chain loss), the calibration parameters were carefully replicated.

We show the effect of symmetrization on a 4-qubit random circuit (Fig. 5a) in eight implementations with varying qubit assignment onto eight ions (Fig. 5b). We model gate miscalibrations as random under-rotations of multiple two-qubit gates fixed per ion pair. We assume an average under-rotation across all qubit pairs causing a similar error for all variants (Fig. 2a). Symmetries $\Gamma$ are represented by eight random qubit assignments $\gamma$. For each qubit assignment, we simulated corresponding inaccurate realizations to obtain vectors $\mathbf{h_{u}}+\delta h_{\Gamma\text{-inv}}+\delta h_{\gamma}$.

In the first case (Fig. 2a), we use only vary qubit assignment between the implementations (Fig. 5b) while in the other case (Fig. 2b), we also replace every fourth XX-gate with a phase-flipped XX^-1 gates with pairs of X-gates thus implementing the same ideal unitary but reversing the effect of under-rotation (Fig. 2b).