A simple proof that anomalous weak values require coherence

The quantum mechanical weak value $A_w=\left\langle \phi|A|\psi \right \rangle / \left\langle \phi | \psi \right\rangle$ of an observable $A$ is a measurable quantity associated with an observable $A$ and pre- and post-selected states $\vert\psi \rangle, \vert \phi \rangle$. Much has been discussed about the meaning and metrological uses of anomalous weak values, lying outside of the range of eigenvalues of $A$. We present a simple proof that anomalous weak values require that the (possibly mixed) pre- and post- selection states have coherence in the eigenbasis of $A$. We also present conditions under which anomalous $A_w$ are witnesses of generalized contextuality, dispensing with the operational weak measurement set-up.Comment: 6 pages, 2 figure

Geometries for universal quantum computation with matchgates

Matchgates are a group of two-qubit gates associated with free fermions. They are classically simulatable if restricted to act between nearest neighbors on a one-dimensional chain, but become universal for quantum computation with longer-range interactions. We describe various alternative geometries with nearest-neighbor interactions that result in universal quantum computation with matchgates only, including subtle departures from the chain. Our results pave the way for new quantum computer architectures that rely solely on the simple interactions associated with matchgates.Comment: 6 pages, 4 figures. Updated version includes an appendix extending one of the result

Inequalities witnessing coherence, nonlocality, and contextuality

Quantum coherence, nonlocality, and contextuality are key resources for quantum advantage in metrology, communication, and computation. We introduce a graph-based approach to derive classicality inequalities that bound local, non-contextual, and coherence-free models, offering a unified description of these seemingly disparate quantum resources. Our approach generalizes recently proposed basis-independent coherence witnesses, and recovers all non-contextuality inequalities of the exclusivity graph approach. Moreover, violations of certain classicality inequalities witness preparation contextuality. We describe an algorithm to find all such classicality inequalities, and use it to analyze some of the simplest scenarios.Comment: 20 pages, 6 figure

Non-stabilizerness and entanglement from cat-state injection

Recently, cat states have been used to heuristically improve the runtime of a classical simulator of quantum circuits based on the diagrammatic ZX-calculus. Here we investigate the use of cat-state injection within the quantum circuit model. We explore a family of cat states, $\left| \mathrm{cat}_m^* \right>$, and describe circuit gadgets using them to concurrently inject non-stabilizerness (also known as magic) and entanglement into any quantum circuit. We provide numerical evidence that cat-state injection does not lead to speed-up in classical simulation. On the other hand, we show that our gadgets can be used to widen the scope of compelling applications of cat states. Specifically, we show how to leverage them to achieve savings in the number of injected qubits, and also to induce scrambling dynamics in otherwise non-entangling Clifford circuits in a controlled manner.Comment: 20 pages, 5 figure

Discrete Wigner functions and quantum computational speedup

In [Phys. Rev. A 70, 062101 (2004)] Gibbons et al. defined a class of discrete Wigner functions W to represent quantum states in a finite Hilbert space dimension d. I characterize a set C_d of states having non-negative W simultaneously in all definitions of W in this class. For d<6 I show C_d is the convex hull of stabilizer states. This supports the conjecture that negativity of W is necessary for exponential speedup in pure-state quantum computation.Comment: 7 pages, 2 figures, RevTeX. v2: clarified discussion on dynamics, added refs., published versio

Extending matchgates into universal quantum computation

Matchgates are a family of two-qubit gates associated with noninteracting fermions. They are classically simulatable if acting only on nearest neighbors, but become universal for quantum computation if we relax this restriction or use SWAP gates [Jozsa and Miyake, Proc. R. Soc. A 464, 3089 (2008)]. We generalize this result by proving that any nonmatchgate parity-preserving unitary is capable of extending the computational power of matchgates into universal quantum computation. We identify the single local invariant of parity-preserving unitaries responsible for this, and discuss related results in the context of fermionic systems.Comment: 9 pages, 2 figure

Quantum circuits to measure scalar spin chirality

The scalar spin chirality is a three-body physical observable that plays an outstanding role both in classical magnetism, characterizing non-co-planar spin textures, and in quantum magnetism, as an order parameter for chiral spin liquids. In quantum information, the scalar spin chirality is a witness of genuine tripartite entanglement. Here we propose an indirect measurement scheme, based on the Hadamard test, to estimate the scalar spin chirality for general quantum states. We apply our method to study chirality in two types of quantum states: generic one-magnon states of a ferromagnet, and the ground state of a model with competing symmetric and antisymmetric exchange. We show a single-shot determination of the scalar chirality is possible for chirality eigenstates, via quantum phase estimation with a single auxiliary qutrit. Our approach provides a unified theory of chirality in classical and quantum magnetism.L.I.R. thanks the New Talents program of Fundação Calouste Gulbenkian for financial support. B.M. acknowledges financial support from Fundação para a Ciência e a Tecnologia (FCT)—Portugal through the Ph.D. scholarship No. SFRH/BD/08444/2020. E.F.G. acknowledges support from FCT via project CEECINST/00062/2018, and from the Digital Horizon Europe project FoQaCiA, GA No. 101070558. J.F.R. acknowledges financial support from FCT (Grant No. PTDC/FIS-MAC/2045/2021), Science National Foundation (Switzerland) Sinergia (Grant Pimag), Generalitat Valenciana funding Prometeo2021/017 and MFA/2022/045, and funding from MICIIN-Spain (Grant No. PID2019-109539GB-C41)