arXiv subjects

Jędrzej Burkat

Publications and source records attributed to Jędrzej Burkat.

5 recordsLinked to original sources

arXiv2026

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

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 restricted model captures decoherence-free subspace computation without access to singlet states. We prove that XQP, as well as its constrained family consisting solely of $\sqrt{\mathrm{SWAP}}$ gates, is universal for quantum computation under polynomial overheads in circuit depth and size. This establishes the universality of the $\sqrt{\mathrm{SWAP}}$ gate on its own under a logical encoding of qubits. We further prove that circuits generated by $\sqrt{\mathrm{SWAP}}$ gates are semi-universal, and thus generate $t$-designs for the uniform distribution over SU(2)-invariant unitaries. Finally, we study additional properties of the XQP model, including its relation to the six-vertex and Potts model in statistical physics. Our findings suggest that XQP circuits are naturally suited to near-term hardware and provide a simple platform for experimental demonstrations of quantum advantage.

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arXiv2026

Exponential de Finetti Theorems for Fermionic Gaussian States

We prove an exponential variant of the Gaussian de Finetti theorem: the subsystems of permutation-invariant, free-fermionic Gaussian states are well approximated by convex combinations of almost-i.i.d. states that are Gaussian on subsets of their parts. Our result provides an error bound between the original state and its approximants that decays exponentially in the number of unconstrained parts, becoming super-exponential when the subsystem under consideration is small. The dependence of our bound on the local Hilbert space dimension is polylogarithmic, giving an exponential improvement over the standard de Finetti theorem of [Nat. Phys. 3, 645-649]. Previous works considered Gaussian-symmetric states, which are supported on the trivial irrep of the tensor matchgate representation. We extend these results to the broader class of Gaussian-invariant states, which contains, for example, i.i.d. copies of single-replica mixed Gaussian states. We show that Gaussian-invariant states are precisely the partial traces of Gaussian-symmetric states on locally enlarged replicas and always admit a purification into a larger Gaussian-symmetric state. As an application, we derive explicit convergence bounds for a hierarchy of convex relaxations of polynomial optimisation over fermionic Gaussian states. For variational energy minimisation, the hierarchy gives lower bounds converging to the global Gaussian minimum. Combined with upper bounds from Gaussian trial states, these bounds quantify how far a variational calculation can be from the best energy attainable within the Gaussian family.

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arXiv2026

Structure, Positivity and Classical Simulability of Kirkwood-Dirac Distributions

Kirkwood-Dirac (KD) quasiprobability distributions are increasingly used across quantum information science, yet their computational significance remains unclear. We classify unitary dynamics that preserve KD positivity and connect this structure to classical simulation. In contrast to the discrete Wigner setting, we show that KD positivity preservation, stochastic evolution of quasiprobabilities, and preservation of total non-positivity do not coincide. We identify three classes of positivity-preserving unitaries: type I gates, which are exactly the KD-stochastic unitaries; type II gates, which are non-stochastic yet preserve total non-positivity by permuting and conjugating KD entries; and, for Fourier-conjugate bases in dimension $d=pq$, type III gates, which preserve KD positivity but can change the total non-positivity of non-real distributions. The classification is complete for Haar-random bases and for Fourier-conjugate dimensions $d=p^k$ and $d=pq$, with $p, q$ distinct primes. Adapting the sampling algorithm of Pashayan et al. [PRL 115, 070501], we simulate all of these positivity-preserving circuits efficiently on KD-positive inputs; type III gates, however, create a sharp distinction between real inputs, which remain efficiently simulable, and non-real inputs, whose sampling overhead can grow exponentially. Consequently, for $d=pq$ no resource theory can both treat the KD total non-positivity as a monotone and admit every efficiently simulable positivity-preserving unitary as a free operation.

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arXiv2026

A Lightweight Protocol for Matchgate Fidelity Estimation

We present a low-depth randomised protocol for estimating the entanglement fidelity between an $n$-qubit matchgate circuit $\mathcal{U}$ and its noisy implementation $\mathcal{E}$. Our method uses a Pauli-Liouville representation adapted to Clifford algebra elements, in which matchgate superoperators acquire a block-diagonal form. This structure enables efficient direct fidelity estimation using only Pauli state preparation and measurement, while avoiding the exponentially costly sampling step required for generic unitary channels. Compared with the protocol of Flammia and Liu [PRL 106, 230501], our algorithm gives an exponential improvement in classical sampling complexity and a multiplicative $1/\sqrt{n}$ reduction in expected shot count for matchgate circuits. The protocol also extends, without asymptotic overhead, to matchgate circuits sandwiched between Clifford circuits. For nearest-neighbour $XY(θ)$ gates and Givens rotations we demonstrate an increase in superoperator sparsity, giving an additional $1/\sqrt{n}$ reduction in expected shot count and, to our knowledge, the first scalable fidelity estimation protocol for these important matchgate subgroups.

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arXiv2025

The Quantum Paldus Transform: Efficient Circuits with Applications

We present the Quantum Paldus Transform: an efficient quantum algorithm for block-diagonalising fermionic, spin-free Hamiltonians in the second quantisation. Our algorithm implements an isometry between the occupation number basis of a fermionic Fock space of $2d$ modes, and the Gelfand-Tsetlin (GT) states spanning irreducible representations of the group $U(d) \times SU(2)$. The latter forms a basis indexed by well-defined values of total particle number $N$, global spin $S$, spin projection $M$, and $U(d)$ GT patterns. This realises the antisymmetric unitary-unitary duality discovered by Howe and developed into the Unitary Group Approach (UGA) for computational chemistry by Paldus and Shavitt in the 1970s. The Paldus transform lends tools from the UGA readily applicable to quantum computational chemistry, leading to maximally sparse representations of spin-free Hamiltonians, efficient preparation of Configuration State Functions, and a direct interpretation of quantum chemistry reduced density matrix elements in terms of $SU(2)$ angular momentum coupling. The transform also enables the encoding of quantum information into novel Decoherence-Free Subsystems for use in communication and error mitigation. Our work can be seen as a generalisation of the quantum Schur transform for the second quantisation, made tractable by the Pauli exclusion principle. Alongside self-contained derivations of the underlying dualities we provide fault-tolerant circuit compilation methods with full gate counts for the Paldus transform, resulting in $\mathcal{O}(d^3)$ Toffoli complexity, where a transform on $50$ spatial orbitals would require a modest $5500$ Toffoli gates. This paves the way for significant advancements in quantum simulation on quantum computers enabled by the UGA paradigm.

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