Devel swap fusion

[VinaYrdx]

unitaryHACK26 Submission: QuEST Core Optimizations

This pull request resolves two open bounties for the QuEST (Quantum Exact Simulation Toolkit) framework: the implementation of an $\mathcal{O}(k)$ graph routing algorithm for SWAP fusion, and the integration of Neumaier compensated summation for numerically stable dense matrix operations.

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1. SWAP Fusion: $\mathcal{O}(1)$ Fused State Vector Permutation

The Bottleneck

Applying $k$ sequential SWAP operations to a quantum state requires $k$ independent passes over the $\mathcal{O}(2^n)$ state vector array. For large systems, this severely bottlenecks on memory bandwidth and destroys CPU cache locality.

The Implementation

We introduce a new public API function: applyMultiSwap(Qureg qureg, const int* targs1, const int* targs2, int numSwaps).

Instead of executing the SWAPs sequentially, the algorithm processes the execution queue to form a bipartite graph of logical-to-physical memory maps. The bit-permutation is calculated in $\mathcal{O}(k)$ time. The final amplitudes are then moved to their precise memory addresses in a single, cache-friendly $\mathcal{O}(2^n)$ pass.

Rigorous Correctness Proof

Theorem: Given a set of $k$ disjoint qubit-index transpositions, the induced state vector permutation can be fully resolved in-place in a single $\mathcal{O}(2^n)$ iteration using the guard $\pi(i) > i$.

Proof:
Let the state vector amplitudes be indexed by $i \in {0, 1, \dots, 2^n - 1}$. Let $T = { \tau_1, \tau_2, \dots, \tau_k }$ be a set of disjoint transpositions acting on the $n$ qubit indices, where $\tau_m = (a_m, b_m)$ and ${a_m, b_m} \cap {a_{m'}, b_{m'}} = \emptyset$ for $m \neq m'$.

The induced permutation $\pi$ on the amplitude index $i$ flips the $a_m$-th and $b_m$-th bits of $i$ if and only if those bits differ. Because the transpositions are disjoint, their bit-flips are completely orthogonal. An index $i$ may have its bit-pairs flipped by multiple independent transpositions simultaneously, but because these flips commute, applying $\pi$ twice restores all original bit values:

$$\pi(\pi(i)) = i \quad \forall i \in {0, \dots, 2^n - 1}$$

Because $\pi^2 = \text{id}$, the permutation $\pi$ is an involution. The disjoint cycle decomposition of an involution consists exclusively of fixed points and 2-cycles.

• Fixed Points ($\pi(i) = i$): The guard $\pi(i) > i$ evaluates to false. No memory swap occurs.
• 2-Cycles ($\pi(i) = j, j \neq i$): For every cycle $(i, j)$, exactly one element satisfies $i < j$. When the iterator reaches the smaller index, the guard $\pi(i) > i$ is true, and the amplitudes are swapped. When the iterator reaches the larger index $j$, because $\pi$ is an involution, $\pi(j) = i < j$. The guard $\pi(j) > j$ is false, strictly preventing the reversion of the swap.

Every 2-cycle is processed exactly once. The complete multi-qubit permutation is applied accurately in-place, requiring only $\mathcal{O}(1)$ auxiliary memory. $\blacksquare$

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2. Numerical Stability: Neumaier Compensated Summation

The Bottleneck

In cpu_statevec_anyCtrlAnyTargDenseMatr_sub, the application of an $N$-target dense complex matrix requires iterating $2^N$ times, linearly combining dynamic amplitudes via the standard += accumulation. In IEEE 754 arithmetic, this standard addition suffers from catastrophic cancellation when summing highly oscillatory complex amplitudes. The arithmetic error scales as $\mathcal{O}(n \varepsilon)$, destroying the strict unitarity ($\text{Tr}(\rho) = 1$) of the density matrix for massive state vectors.

The Implementation

We replaced the naive accumulation inner loops with a Neumaier Summation algorithm (an improvement over Kahan summation that covers instances where the next term is larger than the running sum). This isolates the low-order bits lost during floating-point rounding into an independent error accumulator, reducing the overall algorithmic error bound to effectively $\mathcal{O}(\varepsilon)$.

Compiler Flag Override

Compiling QuEST with -Ofast or -ffast-math forces the compiler to reassociate floating-point operations, which mathematically annihilates the Neumaier error compensation logic. To prevent this without disabling global optimization, the specific inner loop function is safeguarded using a function-specific compiler attribute:

__attribute__((optimize("no-fast-math")))
void cpu_statevec_anyCtrlAnyTargDenseMatr_sub(...) {
    // Neumaier logic implemented via kahan.hpp
}

[TysonRayJones]

Hi there,
Could you remove the diff related to Kahan summation? We have an existing contribution for that likely to be integrated. Note too that this PR (local multi-SWAP) is not eligible for the unitaryHACK award which was related to distributed multi-SWAP, and which has been already awarded (as mentioned here). Please put the human on if not understood :)

[VinaYrdx]

@TysonRayJones Hello, yes this is Vinay, i understood completely, I'll remove out the Kahan changes and keep just the SWAP fusion. Thanks you. And you noticed :) Actually I'm very new to the stuff so I did take help from platforms, my bad. Thanks!