Pure-state tomography with the expectation value of Pauli operators

Ma, Xiongfeng; Jackson, Tyler; Zhou, Hui; Chen, Jianxin; Lu, Dawei; Mazurek, Michael D.; Fisher, Kent; Peng, Xinhua; Kribs, David W.; Resch, Katharina; Ji, Zhengfeng; Zeng, Bei; Laflamme, Raymond

doi:10.1103/physreva.93.032140

Cited by 29 publications

(20 citation statements)

References 28 publications

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“…We have verified, via state tomography, the output state in the control register for the algorithm, achieving a fidelity of around 0.70. For the verification of entanglement generated during the algorithm’s operation, the resource demands of state tomography were circumvented by measuring a much reduced number of Pauli measurements to uniquely identify a quantum state 28 . However, this method is quite specialized and cannot be easily generalized to larger systems.…”

Section: Discussionmentioning

confidence: 99%

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Demonstration of Shor’s factoring algorithm for N $$=$$ 21 on IBM quantum processors

Skosana

Tame

2021

Sci Rep

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We report a proof-of-concept demonstration of a quantum order-finding algorithm for factoring the integer 21. Our demonstration involves the use of a compiled version of the quantum phase estimation routine, and builds upon a previous demonstration. We go beyond this work by using a configuration of approximate Toffoli gates with residual phase shifts, which preserves the functional correctness and allows us to achieve a complete factoring of $$N=21$$ N = 21 . We implemented the algorithm on IBM quantum processors using only five qubits and successfully verified the presence of entanglement between the control and work register qubits, which is a necessary condition for the algorithm’s speedup in general. The techniques we employ may be useful in carrying out Shor’s algorithm for larger integers, or other algorithms in systems with a limited number of noisy qubits.

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Section: Discussionmentioning

confidence: 99%

“…To measure this, we can decompose into 293 Pauli expectations as where are the usual Pauli matrices plus the identity. However, the number of measurements needed to obtain all 293 expectation values can be reduced 28 . This is because the measured probabilities from a measurement of a single Pauli expectation value, i.e.…”

Section: Methodsmentioning

confidence: 99%

Demonstration of Shor’s factoring algorithm for N $$=$$ 21 on IBM quantum processors

Skosana

Tame

2021

Sci Rep

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“…The presented test examples confirm the usefulness of the integer programming approach. Our method can be easily incorporated into other existing tomographic strategies [33][34][35]. Also, it is straightforward to generalize our results to quantum process tomography experiments.…”

Section: Discussionmentioning

confidence: 88%

Optimal design of measurement settings for quantum-state-tomography experiments

Huang

Luo

et al. 2017

Phys. Rev. A

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Quantum state tomography is an indispensable but costly part of many quantum experiments. Typically, it requires measurements to be carried in a number of different settings on a fixed experimental setup. The collected data is often informationally overcomplete, with the amount of information redundancy depending on the particular set of measurement settings chosen. This raises a question about how should one optimally take data so that the number of measurement settings necessary can be reduced. Here, we cast this problem in terms of integer programming. For a given experimental setup, standard integer programming algorithms allow us to find the minimum set of readout operations that can realize a target tomographic task. We apply the method to certain basic and practical state tomographic problems in nuclear magnetic resonance experimental systems. The results show that, considerably less readout operations can be found using our technique than it was by using the previous greedy search strategy. Therefore, our method could be helpful for simplifying measurement schemes so as to minimize the experimental effort.

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“…Let 7], the true value of m 0 (d) is given in [8]. We have m 0 (2, 3, 4, 5, 6, 7) = (4,8,10,16,18,23). We compare this with the value of 3d − 2: (4,7,10,13,16,19).…”

Section: Feasibility Of 3d-2 For Psir-completementioning

confidence: 99%

“…In fact, even four product bases are not enough [11]. Eleven is the minimum number of Pauli operators needed to uniquely determine any two-qubit pure state [23].…”

Section: D=4mentioning

confidence: 99%

Pure state ‘really’ informationally complete with rank-1 POVM

Wang

Shang²

2018

Quantum Inf Process

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What is the minimal number of elements in a rank-1 positiveoperator-valued measure (POVM) which can uniquely determine any pure state in d-dimensional Hilbert space H d ? The known result is that the number is no less than 3d − 2. We show that this lower bound is not tight except for d = 2 or 4. Then we give an upper bound of 4d−3. For d = 2, many rank-1 POVMs with four elements can determine any pure states in H2. For d = 3, we show eight is the minimal number by construction. For d = 4, the minimal number is in the set of {10, 11, 12, 13}. We show that if this number is greater than 10, an unsettled open problem can be solved that three orthonormal bases can not distinguish all pure states in H4. For any dimension d, we construct d + 2k − 2 adaptive rank-1 positive operators for the reconstruction of any unknown pure state in H d , where 1 ≤ k ≤ d.

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Pure-state tomography with the expectation value of Pauli operators

Cited by 29 publications

References 28 publications

Demonstration of Shor’s factoring algorithm for N $$=$$ 21 on IBM quantum processors

Demonstration of Shor’s factoring algorithm for N $$=$$ 21 on IBM quantum processors

Optimal design of measurement settings for quantum-state-tomography experiments

Pure state ‘really’ informationally complete with rank-1 POVM

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