The Power of Power-of-SWAP: Postselected Quantum Computation with the Exchange Interaction

• URL: http://arxiv.org/abs/2603.28527v1
• Date: Mon, 30 Mar 2026 14:53:42 GMT
• Title: The Power of Power-of-SWAP: Postselected Quantum Computation with the Exchange Interaction
• Abstract summary: We introduce Exchange Quantum Polynomial Time (XQP) circuits, which comprise quantum computation using only computational basis SPAM and the isotropic Heisenberg exchange interaction.<n>We show that XQP occupies an intermediate position between BPP and BQP, as its efficient multiplicative-error simulation would collapse the hierarchy to its third level.
• Score: 0.3823356975862005
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We introduce Exchange Quantum Polynomial Time (XQP) circuits, which comprise quantum computation using only computational basis SPAM and the isotropic Heisenberg exchange interaction. Structurally, this sub-universal model captures decoherence-free subspace computation without access to singlet states. We show that XQP occupies an intermediate position between BPP and BQP, as its efficient multiplicative-error simulation would collapse the polynomial hierarchy to its third level. We further provide evidence that additive-error simulation of XQP would enable efficient additive-error simulation of arbitrary BQP computations. Remarkably, the restricted family of XQP circuits consisting solely of $\sqrt{\mathrm{SWAP}}$ gates remains hard to simulate to multiplicative error. We additionally prove that circuits generated by $\sqrt{\mathrm{SWAP}}$ gates are semi-universal, generate $t$-designs for the uniform distribution over $SU(2)$-invariant unitaries, and maximise the entangling power within XQP. Finally, we derive structural results linking computational basis states in XQP to the Gelfand-Tsetlin basis of the symmetric group, and expressing XQP output probabilities as partition functions of the six-vertex and Potts models. Our findings indicate that XQP circuits are naturally suited to near-term hardware and provide a promising platform for experimental demonstrations of quantum computational advantage.

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