Magically, the negativity of the discrete Wigner function is useful

Michael Zurel

An important problem in quantum computation is to characterize the resources required for a computational speedup over classical computation. Veitch et al. showed that one necessary condition for a computational speedup in the model of quantum computation with magic states is that the discrete Wigner function representing the input state of the quantum circuit must take negative values. The amount of negativity in the discrete Wigner function quantifies the complexity of classical simulation of a quantum computation with simulation being efficient if the Wigner function is nonnegative everywhere. In this sense, negativity of the Wigner quasiprobability representation serves as an indicator of quantumness from a computational perspective. However, this result only holds for systems of qudits where the local Hilbert space dimension is odd. The first main result discussed in this thesis relates to a discrete Wigner function suitable for describing quantum computation with magic states defined for any local Hilbert space dimension. When the local Hilbert space dimension is odd it subsumes the standard discrete Wigner function. When the local Hilbert space dimension is even, as a result of state-independent proofs of contextuality among multiqubit (or multi-even-dimensional-qudit) Pauli observables, the phase space over which the Wigner function is defined becomes much larger. However, for systems of qubits, the properties required for simulation of quantum computation with magic states remain. This simulation method effectively extends the result described above for odd-dimensional qudits to qubits. Although in the even-dimensional case the phase space is much larger, points in multiqubit phase space can be characterized and classified and some structure can be imparted on the phase space. The second main result discussed here is a hidden variable model for quantum computation with magic states on qubits. This model is similar in structure to quasiprobability representations like the discrete Wigner function, but unlike those representations the model is capable of representing all elements of any quantum computation---states, operations, and measurements---using only classical probabilities. No negativity is required. This calls into question the role of negativity in quasiprobability representations as an indicator of quantumness for models of quantum computation.

Gunnar G. Bjork, A. B. Klimov, Luis Lorenzo Sánchez-Soto

Victor Veitch

This thesis deals with the question of what resources are necessary and sufficient for quantum computational speedup. In particular, we study what resources are required to promote fault tolerant stabilizer computation to universal quantum computation. In this context we discover a remarkable connection between the possibility of quantum computational speedup and negativity in the discrete Wigner function, which is a particular distinguished quasi-probability representation for quantum theory. This connection allows us to establish a number of important results related to magic state computation, an important model for fault tolerant quantum computation using stabilizer operations supplemented by the ability to prepare noisy non-stabilizer ancilla states. In particular, we resolve in the negative the open problem of whether every non-stabilizer resource suffices to promote computation with stabilizer operations to universal quantum computation. \n \nMoreover, by casting magic state computation as resource theory we are able to quantify how useful ancilla resource states are for quantum computation, which allows us to give bounds on the required resources. In this context we discover that the sum of the negative entries of the discrete Wigner representation of a state is a measure of its usefulness for quantum computation. This gives a precise, quantitative meaning to the negativity of a quasi-probability representation, thereby resolving the 80 year debate as to whether this quantity is a meaningful indicator of quantum behaviour. \n \nWe believe that the techniques we develop here will be widely applicable in quantum theory, particularly in the context of resource theories.

William K. Wootters

Ian Richard C Buckley

We prove that the discrete Wigner function of the canonical $N$-qubit state $|\psi_N\rangle = \mathrm{CZ}{01}\cdot(T\otimes I^{\otimes N-1})\cdot|{+}^{\otimes N}\rangle$ satisfies an exact conservation law for all $N \geq 2$. Three quantities are independent of system size: the total negative Wigner mass $\sum{W<0} W_N(u) = -\sqrt{2}/8$, the total Wigner mass $\sum_u W_N(u) = \cos^2(\pi/8) = (2+\sqrt{2})/4$, and the total variation $\sum_u |W_N(u)| = (1+\sqrt{2})/2$. The conserved quantity $(1+\sqrt{2})/2 = \delta_S/2$, where $\delta_S = 1+\sqrt{2}$ is the silver ratio, reflects the $T$-gate angle $\pi/4$ via the half-angle identity $\cos^2(\pi/8) = (2+\sqrt{2})/4$. The $T$ gate imprints a fixed Wigner defect of magnitude $\sqrt{2}/8$ into phase space; adding $|{+}\rangle$ spectator qubits spreads this defect across $2^{N-2}$ cells but never changes its total. The conservation follows from the factorisation $W_N = W_2 \otimes W_{|+\rangle}^{\otimes N-2}$ and the fact that $\sum_u W_{|+\rangle}(u) = 1$ exactly for any stabiliser state. The conserved constants $\cos^2(\pi/8)$ and $(1+\sqrt{2})/2$ are the tier-boundary constants of the three-tier magic resource theory (Buckley 2026, Hot Logic). The conservation law explains why these constants are fundamental: they are $N$-independent invariants of the entire single-$T$ family, not properties of any specific state. The dark-magic tier is stable under adding spectator qubits. All five quantities are confirmed to machine precision for $N = 2, 3, 4, 5, 6$ by experiment x471b. Keywords: Wigner function · magic states · resource theory · T gate · stabiliser states · discrete Wigner function · silver ratio · conservation law · Clifford hierarchy · quantum computation · Origami ISA · dark magic

Ijaz Durrani