Advantage for Discrete Variational Quantum Algorithms in Circuit Recompilation

TL;DR

This work demonstrates an exponential separation between adaptive and non-adaptive strategies for discrete circuit recompilation by embedding $D$ hidden T gates within layers of partially random unitaries acting on highly entangled quantum data. The authors formulate a quantum analogue of LeadingOnes, define a unimodal but non-separable loss landscape, and show that simple adaptive hill climbing converges in polynomial time while non-adaptive search scales exponentially, even in the presence of realistic shot noise. They analyze landscape properties, concentration behavior, and classical-shadow surrogacy, arguing that adaptive, feedback-driven optimization is essential in this setting. Large-scale numerical tests with rotation-based circuits corroborate scalability and robustness, underscoring potential practical advantages for fault-tolerant recompilation and related dynamical learning tasks in quantum systems.

Abstract

The relative power of quantum algorithms, using an adaptive access to quantum devices, versus classical post-processing methods that rely only on an initial quantum data set, remains the subject of active debate. Here, we present evidence for an exponential separation between adaptive and non-adaptive strategies in a quantum circuit recompilation task. Our construction features compilation problems with loss landscapes for discrete optimization that are unimodal yet non-separable, a structure known in classical optimization to confer exponential advantages to adaptive search. Numerical experiments show that optimization can efficiently uncover hidden circuit structure operating in the regime of volume-law entanglement and high-magic, while non-adaptive approaches are seemingly limited to exhaustive search requiring exponential resources. These results indicate that adaptive access to quantum hardware provides a fundamental advantage.

Figures (11)

• Figure 1: Quantum circuit for T gate puzzle task. Hidden $\mathrm{T}_{q[i]}$ gates are placed on selected qubits $i$ according to a secret bitstring $s = [s_1, s_2, s_3]$, each conjugated by some unitary transformation $V_i$ and layered with circuits $W_i$, representing partially-random unitaries. The positions $q[i]$ are selected at random between $1$ and $n$, and they are known. The resulting quantum state serves as input data for the recompilation task, where the goal is to recover the correct gate placement via discrete optimization. Here we show an example where $q = (3, 1, 6)$ and the solution is $s^*=101$.
• Figure 2: Convergence and scaling of discrete variational protocol. (a) Average loss for hill climbing algorithm shows convergence with each iteration, adaptively approaching the global optimum in 50 instances out of 50. The standard deviation is due to different initial bitstrings $s^{(0)}$ and random unitary realizations. We use $n=D=10$ and $\beta_{\mathrm{W}} = \beta_{\mathrm{V}} = 0.2$, but behavior holds for a broad range of parameters. (b) Scaling for the number of function evaluations until convergence at increasing $D=n$, using $50$ instances and logarithmic y-scale. For adaptive search this follows a quadratic scaling, $n^2/2 - n/4$ (red dashed curve), typical of a unimodal landscape. Error bars show a standard deviation. For non-trivial landscapes, non-adaptive methods are equivalent to random search and follow an exponential scaling, $2^{n}/2$ (blue dot-dashed curve). Shading indicates the interquartile range (25th–75th percentile) across $50$ runs.
• Figure 3: Landscape properties of discrete variational circuit recompilation. (a) Heatmap plot of the fraction of puzzles that are unimodal (0 = all unimodal, 1 = all non-unimodal), shown over six problem instances for $n=D=8$, plotted for different $\beta_{\mathrm{W}} = \beta_{\mathrm{V}}$. The region with significant randomness ($\beta_{\mathrm{W}} = \beta_{\mathrm{V}}>0.15$) shows clear unimodality. (b) Heatmap plot of the fraction of problems that are non-separable (0 = all separable, 1 = all non-separable), shown for the same instances and system sizes as in (a). The landscape remains non-separable for all problems as there are always non-trivial configurations that prevent the separation. As all randomly selected problem instances were also non-monotonic (b) also covers this case.
• Figure 4: Noisy discrete optimization.(a) Loss vs Hamming distance to optimal solution, for $n=D=8$, $\beta_{\mathrm{W}} = \beta_{\mathrm{V}}=0.2$, shows a clear trend (the average value of $\ell_a$ is shown by the purple curve). Relevant properties of the landscape (distance between mean values $\Delta_{\mathrm{S}}$ and gaps $\delta$) are visualized. (b) Properties of the loss landscape that help to explain the success of optimization for landscapes with small gaps between individual loss values. The single local step loss difference $\Delta_S$, which determines the viability of hill climbing, remains constant with $n$. In contrast $\delta$, which does not matter for hill climbing but instead roughly quantifies the loss difference between random landscape points, shrinks with $n$. (c) Hill climbing under finite number of shots, showing an average success rate for the given standard deviation $\sigma$ (defined by number of shots per evaluation). Results are shown for problems with $n=D=6,8,10$, and $\beta_{\mathrm{W}} = \beta_{\mathrm{V}}=0.2$.
• Figure 5: Hardness of puzzle circuits.(a) Entanglement properties of different puzzle circuits for $n = D$, showing the purity of 2-body reduced density matrices after subtracting 1/4. This is equivalent to the squared two-norm distance to the maximally mixed state. Scaling for different $\beta_{\mathrm{W,V}}$ (per individual block) is shown. (b) The Stablizer norm $M$ (a lower bound on the robustness of magic, a non-Cliffordness measure) is shown as a function of $n$ for different $\beta$. Our previous sweet spot of $\beta = 0.2$ is highlighted in bold.