Quantum Dark Magic: Efficiency of Intermediate Non-Stabiliserness

TL;DR

The paper probes how non-stabiliser resources relate to quantum advantage by marrying Stabiliser Rényi Entropies with a geometric, geodesic view of state evolution toward a problem’s solution subspace. It introduces a permutation-robust distance framework via a problem Hamiltonian $H_c$, linking $\langle H_c \rangle$ to geodesic progress, and shows that non-stabiliser consumption can be made more efficient when the evolution is structurally informed (e.g., QAOA) rather than unstructured. Through SAT experiments on small instances, the authors demonstrate that structured state evolutions achieve smoother, more directed progress to the target space and exhibit meaningful correlations between $|\Delta \mathrm{SRE}|$ and distance reductions, in contrast to erratic unstructured evolutions. The work provides a principled toolkit for analyzing and optimizing quantum resources, with implications for algorithm design and fault-tolerant quantum computing. Overall, it suggests that careful integration of non-stabiliser resources with problem structure can enable more reliable quantum advantage.

Abstract

While there is strong evidence for advantages of quantum over classical computation, the repertoire of computational primitives with proven or conjectured quantum advantage remains limited. Despite considerable progress in delineating the quantum-classical divide, the systematic construction of algorithms with quantum advantage remains challenging, which can be attributed to a still incomplete understanding of the sources of quantum computational power. Non-classical behaviour of quantum systems can be characterised, for instance, by intermediate non-stabiliserness , and might be seen as required condition for quantum advantage. Yet, naively equating non-stabiliserness, non-classicality and quantum advantage would be misleading: Even random Haar sampled states that are of doubtful computational use at all exhibit near-maximal non-stabiliserness. Advancing towards systematic quantum advantage calls for a better understanding of the efficient use of non-classical resources like non-stabiliser states. We present an approach to track the behaviour of non-stabiliserness across various algorithms by pairing resource theory of non-stabiliser entropies with the geometry of quantum state evolution, and introduce permutation agnostic distance measures that reveal and quantify non-stabiliser effects previously hidden by a subset of Clifford operations. We find different efficiency in the use of non-stabiliserness for structured and unstructured variational approaches, and show that greater freedom for classical optimisation in quantum-classical methods increases unnecessary non-stabiliser consumption. Our results open new means of analysing the efficient utilisation of quantum resources, and contribute towards the targeted construction of algorithmic quantum advantage.

Figures (7)

• Figure 1: Evolution of the colour representation of the state in quantum circuits. Every vertical slice at $x=i$ represents the colour spectrum of the state after the $i$th gate. The reduced one qubit density matrices are mapped to a hue-saturation-value colour with $\text{hsv}\ab(\braket<P_0>, \braket<P_{+}>, \braket<P_{+i}>)$. Within a vertical slice, they are sorted according to the $\text{hsv}$ tupel.
• Figure 2: Minimal geodesic distance for increasing circuit depths. Non- Clifford computational progress can be observed prior to the final qubit order reversal when all target space permutations $s_0 \ab(\ab[\mathop{\mathrm{\mathcal{T}}}\nolimits])$ (ochre) are considered. As such effects are not visible in the direct distance to the target space $s_0 \ab(\mathop{\mathrm{\mathcal{T}}}\nolimits)$ (black) that neglects permutations, this demonstrates how potential non-stabiliser effects can be masked by non- Clifford- agnostic measures (lines are used to guide the eye and have no significance). The permutation invariant distance $s_0 \ab(\ab[\mathop{\mathrm{\mathcal{T}}}\nolimits])$ is also much more inline with the structural chances observed in the color spectrum representation of the state evolution under the QFT circuit. See \ref{['fig:colorrep']} for a detailed description of the color spectrum representation.
• Figure 3: QFT circuit with four qubits. The dashed box marks the qubit order inversion block of swap gates. Non-stabiliser computations take place before this block, but their computational influence on the geodesic distance is masked by the final qubit reordering.
• Figure 4: Comparison of intermediate geodesic distances $s_0 \ab(\ab[\mathop{\mathrm{\mathcal{T}}}\nolimits])$, non-stabiliserness $\mathop{\mathrm{\text{SRE}}}\nolimits$ and non-stabiliser consumption $\abs{\Delta \mathop{\mathrm{\text{SRE}}}\nolimits}$ between unstructured (\ref{['fig:vqaheatmaps']}) and structured (\ref{['fig:qaoaheatmaps']}) state evolution.
• Figure 5: The $i$-th layer of the hardware efficient ansatz used for unstructured state evolution.

Theorems & Definitions (17)

• Definition 1: see Ref. Leone2022
• Theorem 1: see Leone2022
• proof
• Corollary 1
• proof
• Definition 2
• Remark 1
• Definition 3
• Theorem 2
• proof