Weak Values: Approach through the Clifford and Moyal Algebras

Deformed Clifford algebra and supersymmetric quantum mechanics on a phase space with applications in quantum optics

In order to realize supersymmetric quantum mechanics methods on a four dimensional classical phase-space, the complexified Clifford algebra of this space is extended by deforming it with the Moyal star-product in composing the components of Clifford forms. Two isospectral matrix Hamiltonians having a common bosonic part but different fermionic parts depending on four real-valued phase space functions are obtained. The Hamiltonians are doubly intertwined via matrix-valued functions which are divisors of zero in the resulting Moyal-Clifford algebra. Two illustrative examples corresponding to Jay…

Approximate stabilizer rank and improved weak simulation of Clifford-dominated circuits for qudits

Bravyi and Gosset recently gave classical simulation algorithms for quantum circuits dominated by Clifford operations. These algorithms scale exponentially with the number of T-gate in the circuit, but polynomially in the number of qubits and Clifford operations. Here we extend their algorithm to qudits of odd prime dimensions. We generalize their approximate stabilizer rank method for weak simulation to qudits and obtain the scaling of the approximate stabilizer rank with the number of single-qudit magic states. We also relate the canonical form of qudit stabilizer states to Gauss sum evaluat…

Quantum Trajectories: Dirac, Moyal and Bohm

We recall Dirac's early proposals to develop a description of quantum phenomena in terms of a non-commutative algebra in which he suggested a way to construct what he called `quantum trajectories'. Generalising these ideas, we show how they are related to weak values and explore their use in the experimental construction of quantum trajectories. We discuss covering spaces which play an essential role in accounting for the `wave' properties of quantum particles. We briefly point out how new mathematical techniques take us beyond Hilbert space and into a deeper structure which connects with the …

射影误差模型：稳定码、克利夫顿码和弱稳定码

Projective error models: Stabilizer codes, Clifford codes, and weak stabilizer codes

本文通过射影误差模型研究稳定码、克利夫顿码和弱稳定码的数学结构，并给出了非稳定码克利夫顿码的构造方法。

随机监测量子电路中的不变测度与弱魔法注入渐近行为

Invariant Measures and Weak-Magic-Injection Asymptotics in Random Monitored Quantum Circuits

本文为随机监测量子电路建立严格数学框架，证明平稳分布的存在唯一性，并解析了弱魔法注入极限下稳态魔法的领先渐近行为，揭示了不同资源度量在稳定子层附近的局部几何差异。

How the Weak Variance of Momentum Can Turn Out to be Negative

Weak values are average quantities,therefore investigating their associated variance is crucial in understanding their place in quantum mechanics. We develop the concept of a position-postselected weak variance of momentum as cohesively as possible, building primarily on material from Moyal (Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, Cambridge, 1949) and Sonego (Found Phys 21(10):1135, 1991) . The weak variance is defined in terms of the Wigner function, using a standard construction from probability theory. We show this corresponds to a measur…