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

Jędrzej Burkat, Sergii Strelchuk, Michał Studziński

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.

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

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 gates remains hard to simulate to multiplicative error. We additionally prove that circuits generated by gates are semi-universal, generate -designs for the uniform distribution over -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.

Paper Structure

This paper contains 11 sections, 15 theorems, 117 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. The Postselected Phase Gadget
  4. Multiplicative Weak Simulation of XQP Circuits Implies Collapse of the Polynomial Hierarchy
  5. Additive Weak Simulation of XQP Circuits is Unlikely
  6. Semi-Universality of XQP(pi/4) Circuits
  7. The Entangling Power of Exchange Gates
  8. Characterisation of Irrep Structure for XQP States
  9. XQP Circuits as Six-Vertex and Potts Models
  10. Open Problems
  11. Alternative Proof of Theorem 1

Key Result

Theorem 1

$\mathsf{PostXQP(\pi/4)} = \mathsf{PostXQP} = \mathsf{PostBQP}$

Figures (5)

  • Figure 1: Quantum Circuit for the Phase Gadget. Upon successful postselection on the ancilla measurements, the gadget implements a phase gate $e^{i \theta Z }$ on the input state. The success probability is given by $p(\text{success}) = \cos^2 \theta$.
  • Figure 2: Six-vertex Model Weights. Each vertex contains the same number of incoming and outgoing arrows (the ice rule). We can treat the six possible vertex arrangements in the top row as the trajectories of $|0 \rangle$ (thin lines corresponding to upwards arrows) and $|1 \rangle$ (thick lines corresponding to downwards arrows) by rotating $45^{\circ}$ in the anti-clockwise direction. For the $U(\theta) = \cos \theta \cdot \mathbf{1} + i \sin \theta \cdot E_{ij}$ exchange gates, the corresponding weights are given by $a_1 = a_2 = e^{i \theta}$, $b_1 = b_2 = i \sin \theta$, and $c_1 = c_2 = \cos \theta$.
  • Figure 3: Domain Wall Boundary Conditions (DWBC).Left: An $\mathsf{XQP}$ circuit of DWBC type, where each gate is an exchange interaction $U(\arctan (\lambda_i - \nu_j))$. Right: The corresponding six-vertex model with DWBC, where a vertex connecting lines with spectral parameters $\lambda_i$ and $\nu_j$ has weights $a(\lambda_i, \nu_j) = 1 + i(\lambda_i - \nu_j)$, $b(\lambda_i, \nu_j) = i (\lambda_i - \nu_j)$, and $c(\lambda_i, \nu_j) = 1$.
  • Figure 4: The Square Potts Model. White dots and dashed lines represent the vertices and edges of the square Potts model on $\mathcal{L}$. Black dots and full lines represent the vertices and edges of the equivalent modified six-vertex model on the medial lattice $\mathcal{L}'$. Black dots on dashed lines are the internal vertices, and all other dots are the external vertices. Left: The $3 \times 3$ square Potts model. Right: The $2 \times 2$ Potts model with cylindrical boundary conditions along the horizontal direction (external vertices on the sides are connected by a red, dotted 'seam'). Figure reproduced from Baxter_1982.
  • Figure 5: Quantum Circuits for the Square Potts Model. The probability amplitudes obtained from the above circuits are proportional to the partition function of the square Potts model (Figure \ref{['fig:potts']}). The $U_S$ gate acts as $U_S | 01 \rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)$, allowing for singlet state preparation and measurement. When boundary conditions are periodic, additional $U(\theta_h)$ gates are implemented between the first and last qubit at each 'horizontal' layer.

Theorems & Definitions (45)

  • Definition 1: Decoherence-Free Subspace Computation Kempe_2001
  • Definition 2: Exchange Quantum Polynomial Time Circuits
  • Definition 3
  • Definition 4: Postselected Exchange Quantum Polynomial Time
  • Remark 1
  • proof
  • Definition 5: Schur-Weyl Duality
  • Theorem 1
  • proof
  • Corollary 1
  • ...and 35 more