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arXiv:1401.5964 · Physical Review A

Bounds on quantum process fidelity from minimum required number of quantum state fidelity measurements

Jaromı́r Fiurášek, Michal Sedlák

Published Jan 22, 2014• 7 citations
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Abstract

To certify that an experimentally implemented quantum transformation is a certain unitary operation $U$ on a $d$-dimensional Hilbert space, it suffices to determine fidelities of output states for $d+1$ suitably chosen pure input states [Reich et al., Phys. Rev. A 88, 042309 (2013)]. The set of these $d+1$ probe states can consist of $d$ orthogonal states that form a basis and one additional state which is a balanced superposition of all $d$ basis states. Here we provide an analytical lower bound on quantum process fidelity for two-qubit quantum gates which results from the knowledge of average state fidelity for the basis states and the fidelity of the superposition state. We compare this bound with the Hofmann bound that is based on knowledge of average state fidelities for two mutually unbiased bases. We also discuss the possible extension of our findings to $N$-qubit operations.

DOI
10.1103/physreva.89.012323

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To certify that an experimentally implemented quantum transformation is a certain unitary operation UUU on a ddd-dimensional Hilbert space, it suffices to determine fidelities of output states for d+1d+1d+1 suitably chosen pure input states [Reich et al., Phys. Rev. A 88, 042309 (2013)]. The set of these d+1d+1d+1 probe states can consist of ddd orthogonal states that form a basis and one additional state which is a balanced superposition of all ddd basis states. Here we provide an analytical lower bound on quantum process fidelity for two-qubit quantum gates which results from the knowledge of average state fidelity for the basis states and the fidelity of the superposition state. We compare this bound with the Hofmann bound that is based on knowledge of average state fidelities for two mutually unbiased bases. We also discuss the possible extension of our findings to NNN-qubit operations.
We propose and investigate bounds on the quantum process fidelity of quantum filters, i.e., probabilistic quantum operations represented by a single Kraus operator KKK. These bounds generalize the Hofmann bounds on the quantum process fidelity of unitary operations [H. F. Hofmann, Phys. Rev. Lett. 94, 160504 (2005)] and are based on probing the quantum filter with pure states forming two mutually unbiased bases. Determination of these bounds therefore requires far fewer measurements than full quantum process tomography. We find that it is particularly suitable to construct one of the probe bases from the right eigenstates of KKK, because in this case the bounds are tight in the sense that if the actual filter coincides with the ideal one, then both the lower and the upper bounds are equal to 1. We theoretically investigate the application of these bounds to a two-qubit optical quantum filter formed by the interference of two photons on a partially polarizing beam splitter. For an experimentally convenient choice of factorized input states and measurements we study the tightness of the bounds. We show that more stringent bounds can be obtained by more sophisticated processing of the data using convex optimization and we compare our methods for different choices of the input probe states.
We investigate the problem of bounding the quantum process fidelity given bounds on the fidelities between target states and the action of a process on a set of pure input states. We formulate the problem as a semidefinite program and prove convexity of the minimum process fidelity as a function of the errors on the output states. We characterize the conditions required to uniquely determine a process in the case of no errors and derive a lower bound on its fidelity in the limit of small errors for any set of input states satisfying these conditions. We then consider sets of input states whose one-dimensional projectors form a symmetric positive operator-valued measure (POVM). We prove that for such sets the minimum fidelity is bounded by a linear function of the average output state error. A symmetric POVM with minimal number of elements contains d+1d+1d+1 states, where ddd is the Hilbert space dimension. Our bounds applied to such states provide an efficient method for estimating the process fidelity without the use of full process tomography.
For target unitary operations which preserve the basis of measurement operators, the average fidelity of the corresponding N -qubit gate can be determined efficiently. That is, the number of required experiments is independent of system size and the classical computational resources scale only polynomially in the number N of qubits. Here we address the question of how to optimally choose the measurement basis for fidelity estimation when replacing two-level qubits by d -level qudits. We define optimality in terms of the maximal number of unitaries that preserve the measurement basis. Our definition allows us to construct the optimal measurement basis in terms of their spectra and eigenbases: the measurement operators are unitaries with d -nary spectrum and partition into Abelian groups whose eigenbases are mutually unbiased.
We propose an efficiently measurable lower bound on quantum process fidelity of N-qubit controlled-Z gates. This bound is determined by average output state fidelities for N partially conjugate product bases. A distinct advantage of our approach is that only fidelities with product states need to be measured while keeping the total number of measurements much smaller than what is necessary for full quantum process tomography. As an application, we use this method to experimentally estimate quantum process fidelity F of a three-qubit linear optical quantum Toffoli gate and we find that F≥0.83. We also demonstrate the entangling capability of the gate by preparing Greenberger-Horne-Zeilinger-type three-qubit entangled states from input product states.
We define scalable benchmarks for two important quantum subroutines, namely the quantum Fourier transform (QFT) and the multi-controlled-Z gate. Our benchmarks are based on applying the subroutines to a set of states that are sufficient for lower-bounding the process fidelity from the set of measured output state fidelities. By randomly sampling from the set of input states, the lower boundonthe process fidelity can be estimated. We implement the QFT and multi-controlled-Z circuits with up to 20 and 12 qubits, respectively, on the Quantinuum H1-1 quantum computer. Our circuits make use of parameterized-angle entangling gates, which for these circuits reduces the number of two-qubit gates significantly.
We derive an algebraic framework which identifies the minimal information required to assess how well a quantum device implements a desired quantum operation. Our approach is based on characterizing only the unitary part of an open system's evolution. We show that a reduced set of input states is sufficient to estimate the average fidelity of a quantum gate, avoiding a sampling over the full Liouville space. Surprisingly, the minimal set consists of only two input states, independent of the Hilbert-space dimension. The minimal set is, however, impractical for device characterization, since one of the states is a totally mixed thermal state and extracting bounds for the average fidelity is impossible. We therefore present two further reduced sets of input states that allow for, respectively, numerical and analytical bounds on the average fidelity.