Dynamically corrected gates suppressing spatiotemporal error correlations as measured by randomized benchmarking

Quantum error correction provides a path to large-scale quantum computers, but is built on challenging assumptions about the characteristics of the underlying errors. In particular, the mathematical assumption of statistically independent errors in quantum logic operations is at odds with realistic environments where error sources may exhibit strong temporal and spatial correlations. We present experiments using trapped ions to demonstrate that the use of dynamically corrected gates (DCGs), generally considered for the reduction of error magnitudes, can also suppress error correlations in space and time throughout quantum circuits. We present a first-principles analysis of the manifestation of error correlations in randomized benchmarking and validate this model through experiments performed using engineered errors. We find that standard DCGs can reduce error correlations by $\sim 50\times$ while increasing the magnitude of uncorrelated errors by a factor scaling linearly with the extended DCG duration compared to a primitive gate. We then demonstrate that the correlation characteristics of intrinsic errors in our system are modified by the use of DCGs, consistent with a picture in which DCGs whiten the effective error spectrum induced by external noise.

Figure 1

Translation of noise correlations to error correlations in quantum circuits. (a) A single operation applied to a qubit in the presence of noise ${\tilde{U}}_j$ can be decomposed into an error operator ${\widehat{\mathrm{\Lambda }}}_j$ and the target operation ${\widehat{U}}_j$. Bloch spheres schematically illustrate the effect of an imperfect $\pi$ rotation about the $x$ axis acting on input state $|1\rangle$, with dark shading indicating an overrotation error. (b) Noise (red line) exhibiting nonzero temporal correlation of length ${\mathcal{M}}_{\mathrm{n}}=3$, quantized in units of gate operations, acts on a quantum circuit composed of sequentially applied unitary operations. The resultant errors accumulate and lead to a noisy effective operator ${\tilde{U}}_{\mathrm{eff}}$, whose effect is determined through a projective measurement at the end of the circuit. (c) Translation of correlations in a noise process to correlations in the magnitude of the circuit error vector $\parallel {\mathit{\epsilon }}_j\parallel$. The error vector for each gate of a randomly composed sequence of 1000 primitive gates under a noise process with noise correlation length ${\mathcal{M}}_{\mathrm{n}}$ is calculated and the autocorrelation function of the magnitude of the error vector $\mathbb{E}[\parallel {\mathit{\epsilon }}_{j_1}\parallel \parallel {\mathit{\epsilon }}_{j_2}\parallel ]$ is shown for the first 100 gates. Random walks are shown for the extreme error correlation cases (d) ${\mathcal{M}}_{\varepsilon }=1$ (uncorrelated) and (e) ${\mathcal{M}}_{\varepsilon }=J$ (fully correlated). Final walk displacements of eight sequences, each with 1000 error realizations, are shown along with the full walk for a single sequence that is common between the two cases.

Figure 2

Signatures of error correlations in randomized benchmarking sequences. (a)–(c) Distribution of measured survival probabilities for $k=50$ randomly composed sequences averaged over $n=5$, 25, and 100 noise realizations drawn from $\delta \sim \mathcal{N}(0,{\rho }^2=2\times {10}^{-3})$ for both maximally correlated ${\mathcal{M}}_{\mathrm{n}}=J$ (gray) and uncorrelated (red) engineered noise processes. Uncorrelated noise possesses a $\pi /2$ bandwidth, i.e., noise values change with a rate set at the inverse of the duration of a primitive $\pi /2$ rotation, and hence can take one or multiple values in a gate (${\mathcal{M}}_{\mathrm{n}}\le 1$). Solid lines are normalized $\mathrm{\Gamma }$ distributions plotted with no free parameters. (d) Scaling of cumulatively noise-averaged histogram variances ${\mathbb{V}}_{\mathrm{k}}^{\left(n\right)}\equiv {\mathbb{V}}_{\mathrm{k}}\left[{\langle P\left(|1\rangle \right)\rangle }_n\right]$. Trajectories correspond to different orderings of noise realizations with dotted lines representing the mean of 1000 reorderings and solid lines are theoretical predictions with no free parameters (see the text). Vertical dashed lines indicate the values of $n$ used in (a)–(c).

Figure 3

Suppression of error correlations using dynamically corrected gates. (a) The first-order generalized filter-transfer function for dephasing noise of a primitive operation $G_z^{\left(1\right)}(\omega ,T_j)$ and the noise spectrum [here ${\beta }_z\left(\omega \right)\propto 1/\omega ]$ combine to produce an effective error spectrum $E(\omega ,T_j)$ for a single gate. (b) The modified filter functions for first-order DCGs scale as $\omega$ at low frequencies, which results in a whitening of $E(\omega ,T_j)$ relative to the input noise spectrum. (c) and (d) Variance scaling with $n$ for primitive (gray) gates and WAMF (orange), CORPSE (blue), and BB1 (green) DCGs all subjected to noise with both correlated and uncorrelated components. In (c) detuning noise is engineered with strength ${\delta }_C\sim \mathcal{N}(0,2\times {10}^{-3})$ and ${\delta }_U\sim \mathcal{N}(0,5\times {10}^{-4})$ and in (d) amplitude noise is engineered with strength ${\delta }_C\sim \mathcal{N}(0,9\times {10}^{-4})$ and ${\delta }_U\sim \mathcal{N}(0,2\times {10}^{-4})$. Dotted lines are means of 1000 trajectories randomized over noise realizations and solid lines for the DCGs are theoretical fits from Table 1 to the mean with the values of ${\sigma }_U^2$ and ${\sigma }_C^2$ allowed to vary. Black solid lines for primitive gates are derived from the same theory with no free parameters. As with Fig. 2, all data are measured for $k=50$ sequences of length $J=100$ with $n=200$ noise realizations and $r=220$ repetitions.

Figure 4

Suppression of error correlations using DCGs under noise with varying ${\mathcal{M}}_{\mathrm{n}}$. Variance scaling of $k=20$ sequences with noise averaging is shown for (a) primitive and (b) CORPSE gates. Traces are normalized to the initial mean variance for each applied noise case. Engineered noise is composed of an uncorrelated component (${\mathcal{M}}_{\mathrm{n}}\le 1$) and a block correlated component of length ${\mathcal{M}}_{\mathrm{n}}$ that is varied from fully correlated (${\mathcal{M}}_{\mathrm{n}}=J$) to uncorrelated (${\mathcal{M}}_{\mathrm{n}}=1$) in units of virtual gates. Dotted lines are means of 1000 randomized trajectories. (c) Ratio of initial to final variance in (a) and (b) as ${\mathcal{M}}_{\mathrm{n}}$ is varied for primitive (black) and CORPSE (blue) gates. The dotted line marks the ratio at which CORPSE gates saturate and the dashed vertical line indicates the value of ${\mathcal{M}}_{\mathrm{n}}$ where this ratio crosses the scaling trend for primitive gates. Error bars calculated from the standard error of the mean of the 200 initial values of variance and normalized by the fully noise-averaged variance are smaller than point size.

Figure 5

Intrinsic errors in a five-qubit chain. (a) Variance over noise-averaged sequence survival probabilities for five qubits using $k=60$ sequences of length $J=500$, averaged over repetitions $r$, up to $r=500$. Each trajectory is produced by shuffling the order of repetitions used in the graph to avoid bias, dotted lines indicate the means of 1000 trajectory randomizations, and solid lines are fits where the correlated and uncorrelated error strengths were free to vary. The correlated error strengths ${\sigma }_C^2$ are $\{1.2,1.5,1.9,2.4,2.7\}\times {10}^{-4}$ from qubit 1 to 5 for the primitive gates and $\{2.3,2.5,1.1,2.2,2.3\}\times {10}^{-5}$ for the BB1 gates. The uncorrelated error strengths ${\sigma }_U^2$ are $\{7.5,8.1,8.5,8.6,8.7\}\times {10}^{-4}$ from qubit 1 to 5 for the primitive gates and $\{6.5,6.5,6.3,6.6,6.5\}\times {10}^{-4}$ for the BB1 gates. The insert shows the EMCCD image of a five-ion chain, spaced over $\sim 30\mu \mathrm{m}$. The control field amplitude and frequency are calibrated with respect to the highlighted leftmost ion. (b) Pairwise cross-correlation coefficients between the five-qubit survival probabilities for primitive gates (left) and BB1 DCGs (right), revealing a $\sim 50\%$ reduction in the correlations between qubit errors when using DCGs.

Figure 6

Single-qubit randomized benchmarking. Randomized benchmarking is performed on a single qubit using a total of 300 sequences composed of primitive gates, with 50 sequences each of length $J=2$, 25, 50, 100, 200, and 500. Each sequence was repeated $r=500$ times to reduce quantum projection noise. Black markers represent individual sequence survival probabilities, red crosses indicate the mean survival probabilities for each sequence length, and the solid red line is a fit to the means to extract the average error per gate.

Figure 7

Quantum projection noise limits for measured survival probabilities with the CORPSE DCG. Comparison of mean CORPSE variance scaling from Fig. 3 (blue) to QPN variance limits given by $p(1-p)/r$. The dashed line is the worst case QPN for $r=220$ when $p=0.5$. Black lines show additional QPN limits where, for each $n, p(1-p)/r$ is calculated for 100 randomizations of noise realizations. The lower line scaling as $1/n$ is divided by $n\times r$ rather than $r$.