We report experimental verification of the commutation relation for Pauli spin operators using quantum interference of the single-photon polarization state. By superposing the quantum operations $σ_z σ_x$ and $σ_x σ_z$ on a single-photon polarization state, we have experimentally implemented the commutator, $[σ_{z}, σ_{x}]$, and the anticommutator, $\{σ_{z}, σ_{x}\}$, and have demonstrated the relative phase factor of $π$ between $σ_z σ_x$ and $σ_x σ_z$ operations. The experimental quantum operation corresponding to the commutator, $[σ_{z}, σ_{x}]=kσ_y$, showed process fidelity of 0.94 compared to the ideal $σ_y$ operation and $|k|$ is determined to be $2.12\pm0.18$.
Using dyadic representations elaborated from vectors of Jones, and calculating relations of anti-commutation of these tensorial forms, we obtain in shape explicit the Pauli spin matrices.
The quantum weak value draws many attentions recently from theoretical curiosity to experimental applications. Now we design an unusual weak measuring procedure as the pre-selection, mid-selection and post-selection to study the correlation function of two weak values, which we called the weak correlation function. In this paper, we proposed an weak measurement experiment to measure the canonical commutator $[\hat{x},\hat{p}]=i\hbar$ in quantum mechanics. Furthurmore, we found the intriguing equivalence between the canonical commutation relation and Riemann hypothesis, and then obtained the we…
Experimental Demonstration of Commutation Relations Using Intensity Correlations
Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学）；Max-Planck-Institut für die Physik des Lichts（马克斯·普朗克光科学研究所）；Universidad Complutense（康普卢顿塞大学）；Chapman University（查普曼大学）；University of Ottawa（渥太华大学）
Many-body quantum states, as a matter of fact, are extremely essential to solve certain mathematical problems or simulate quantum systems in measurement-based quantum computation. However, how to verify large scale quantum states, such as hypergraph states, is an exceedingly hard task for multi-body quantum systems. Here, we propose a novel fault-tolerant solution for verification of colorable hypergraph states by using stabilizer test. Compared with the adaptive stabilizer test, our protocol is dramatically facilitating by making only Pauli-X and Pauli-Z measurement. As to appliance, it will …
Statistical verification of a quantum state aims to certify whether a given unknown state is close to the target state with confidence. So far, sample-optimal verification protocols based on local measurements have been found only for disparate groups of states: bipartite pure states, GHZ states, and antisymmetric basis states. In this work, we investigate systematically optimal verification of entangled stabilizer states using Pauli measurements. First, we provide a lower bound on the sample complexity of any verification protocol based on separable measurements, which is independent of the n…
This paper considers a class of open quantum systems with an algebraic structure of dynamic variables, including the Pauli matrices for finite-level systems as a particular case. The Hamiltonian and the operators of coupling of the system to the external bosonic fields depend linearly on the system variables. The fields are represented by quantum Wiener processes which drive the system dynamics in the form of a quasilinear Hudson-Parthasarathy quantum stochastic differential equation whose drift vector and dispersion matrix are affine and linear functions of the system variables. This quasilin…
Based on Lorentz invariance and Born reciprocity invariance, the canonical quantization of Special Relativity (SR) is shown to provide a unified origin for: i) the complex vector space formulation of Quantum Mechanics (QM); ii) the momentum and space commutation relations and the corresponding representations; iii) the Dirac Hamiltonian in the formulation of Relativistic Quantum Mechanics (RQM); iv) the existence of a self adjoint Time Operator that circumvents Pauli's objection.
Measurement-based quantum computing is one of the most promising quantum computing models. Although various universal resource states have been proposed so far, it was open whether only two Pauli bases are enough for both of universal measurement-based quantum computing and its verification. In this paper, we construct a universal hypergraph state that only requires $X$ and $Z$-basis measurements for universal measurement-based quantum computing. We also show that universal measurement-based quantum computing on our hypergraph state can be verified in polynomial time using only $X$ and $Z$-bas…
Pauli's Theorem and Quantum Canonical Pairs: The Consistency Of a Bounded, Self-Adjoint Time Operator Canonically Conjugate to a Hamiltonian with Non-empty Point Spectrum
Noisy Intermediate-Scale Quantum (NISQ) algorithms, which run on noisy quantum computers should be carefully designed to boost the output state fidelity. While several compilation approaches have been proposed to minimize circuit errors, they often omit the detailed circuit structure information that does not affect the circuit depth or the gate count. In the presence of spatial variation in the error rate of the quantum gates, adjusting the circuit structure can play a major role in mitigating errors. In this paper, we exploit the freedom of gate reordering based on the commutation rules to s…
Verification is a task to check whether a given quantum state is close to an ideal state or not. In this paper, we show that a variety of many-qubit quantum states can be verified with only sequential single-qubit measurements of Pauli operators. First, we introduce a protocol for verifying ground states of Hamiltonians. We next explain how to verify quantum states generated by a certain class of quantum circuits. We finally propose an adaptive test of stabilizers that enables the verification of all polynomial-time-generated hypergraph states, which include output states of the Bremner-Montan…
Key Laboratory of System Software (Chinese Academy of Sciences), Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所系统软件重点实验室）；University of Chinese Academy of Sciences（中国科学院大学）；Leiden University（莱顿大学）；University of Edinburgh（爱丁堡大学）
Efficient verification of the functioning of quantum devices is a key to the development of quantum technologies, but is a daunting task as the system size increases. Here we propose a simple and general framework for verifying unitary transformations that can be applied to both individual quantum gates and gate sets, including quantum circuits. This framework enables efficient verification of many important unitary transformations, including but not limited to all bipartite unitaries, Clifford unitaries, generalized controlled-$Z$ gates, generalized controlled-NOT gates, the controlled-SWAP g…
Hypergraph states are generalizations of graph states where controlled-$Z$ gates on edges are replaced with generalized controlled-$Z$ gates on hyperedges. Hypergraph states have several advantages over graph states. For example, certain hypergraph states, such as the Union Jack states, are universal resource states for measurement-based quantum computing with only Pauli measurements, while graph state measurement-based quantum computing needs non-Clifford basis measurements. Furthermore, it is impossible to classically efficiently sample measurement results on hypergraph states with a constan…
The finite entropy of black holes suggests that local regions of spacetime are described by finite-dimensional factors of Hilbert space, in contrast with the infinite-dimensional Hilbert spaces of quantum field theory. With this in mind, we explore how to cast finite-dimensional quantum mechanics in a form that matches naturally onto the smooth case, especially the recovery of conjugate position/momentum variables, in the limit of large Hilbert-space dimension. A natural tool for this task are the Generalized Pauli operators (GPO). Based on an exponential form of Heisenberg's canonical commuta…
Among various multipartite entangled states, Dicke states stand out because their entanglement is maximally persistent and robust under particle losses. Although much attention has been attracted for their potential applications in quantum information processing and foundational studies, the characterization of Dicke states remains as a challenging task in experiments. Here, we propose efficient and practical protocols for verifying arbitrary $n$-qubit Dicke states in both adaptive and nonadaptive ways. Our protocols require only two distinct settings based on Pauli measurements besides permut…
While quantum speed-up in solving certain decision problems by a fault-tolerant universal quantum computer has been promised, a timely research interest includes how far one can reduce the resource requirement to demonstrate a provable advantage in quantum devices without demanding quantum error correction, which is crucial for prolonging the coherence time of qubits. We propose a model device made of locally-interacting multiple qubits, designed such that simultaneous single-qubit measurements on it can output probability distributions whose average-case sampling is classically intractable, u…
QCIVET: A Quantum--Classical Pipeline Integrity Framework with Contract-Based Subtype Verification and Hash-Chained Audit Traces
Copenhagen School of Business (EK – Erhvervsakademi København)（哥本哈根商学院）；Department of Computer Engineering, Istanbul Gelisim University（伊斯坦布尔吉利什姆大学计算机工程系）
Quantum resource analysis is crucial for designing quantum circuits as well as assessing the viability of arbitrary (error-corrected) quantum computations. To this end, we introduce QUANTIFY, which is an open-source framework for the quantitative analysis of quantum circuits. It is based on Google Cirq and is developed with Clifford+T circuits in mind, and it includes the necessary methods to handle Toffoli+H and more generalised controlled quantum gates, too. Key features of QUANTIFY include: (1) analysis and optimisation methods which are compatible with the surface code, (2) choice between …
