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Qutrit randomized benchmarking
A. Morvan, V. V. Ramasesh, M. S. Blok, J. M. Kreikebaum, K. O'Brien, L. Chen, B. K. Mitchell, R. K. Naik, D. I. Santiago, I. Siddiqi
TL;DR
Qutrit processors need benchmarking protocols suited to their higher-dimensional state spaces, since qubit-like RB has limitations for characterizing qutrit gates. This work extends randomized and cycle benchmarking to qutrits and demonstrates them on a superconducting processor, reporting single-qutrit errors as low as 2.38×10^-3 and a two-qutrit CSUM process fidelity of 0.82.
Problem
Qubit-like randomized benchmarking yields separate subspace results and does not support interleaved benchmarking of individual gates, motivating protocols suited to qutrits.
Source: p. 2Method
The paper extends randomized benchmarking and cycle benchmarking to qutrits, using interleaved and simultaneous RB alongside cycle benchmarking on a superconducting qutrit processor.
Source: p. 1Results
Single-qutrit randomized-benchmarking numbers reach 2.38×10^-3, while the two-qutrit CSUM process fidelity is 0.82.
Source: pp. 1, 5Takeaways & Limitations
The protocols enable qutrit processors to be meaningfully compared with qubit-based counterparts.
Source: p. 5
Abstract
from arXiv · showhide
Ternary quantum processors offer significant computational advantages over conventional qubit technologies, leveraging the encoding and processing of quantum information in qutrits (three-level systems). To evaluate and compare the performance of such emerging quantum hardware it is essential to have robust benchmarking methods suitable for a higher-dimensional Hilbert space. We demonstrate extensions of industry standard Randomized Benchmarking (RB) protocols, developed and used extensively for qubits, suitable for ternary quantum logic. Using a superconducting five-qutrit processor, we find a single-qutrit gate infidelity as low as $2.38 \times 10^{-3}$. Through interleaved RB, we find that this qutrit gate error is largely limited by the native (qubit-like) gate fidelity, and employ simultaneous RB to fully characterize cross-talk errors. Finally, we apply cycle benchmarking to a two-qutrit CSUM gate and obtain a two-qutrit process fidelity of $0.82$. Our results demonstrate a RB-based tool to characterize the obtain overall performance of a qutrit processor, and a general approach to diagnose control errors in future qudit hardware.
Supplementary Material: Qutrit randomized benchmarking
The supplied passage identifies the document as arXiv:2008.09134v1 in quant-ph, dated 20 August 2020.
- The document is identified as arXiv:2008.09134v1 in quant-ph, dated 20 August 2020.Source: Supplementary Material: Qutrit… · p. 7
SINGLE QUTRIT GATE DECOMPOSITION
Single-qutrit gates are constructed from rotations in the {|0⟩, |1⟩} and {|1⟩, |2⟩} subspaces. The material describes ZXZXZ and cos-sin decompositions, including an example for the qutrit Hadamard.
SINGLE QUTRIT GATE DECOMPOSITION
- The native qutrit gateset consists of two-level rotations in the {|0⟩, |1⟩} and {|1⟩, |2⟩} subspaces.The (01) or (12) subscript identifies the rotation's subspace.Source: SINGLE QUTRIT GATE DECOMPOSITION · p. 7
- Arbitrary single-qutrit unitaries are decomposed into rotations within two-level subspaces using a cos-sin decomposition.Each subspace rotation uses the ZXZXZ decomposition.Source: SINGLE QUTRIT GATE DECOMPOSITION · p. 7
- An alternative single-qutrit decomposition uses two rotations in the (01) subspace and one in the (12) subspace by relabelling the matrix entries.This alternative was noted during manuscript preparation.Source: SINGLE QUTRIT GATE DECOMPOSITION · p. 7
- A qutrit Hadamard decomposition is given, with α further decomposable using virtual Z gates.The diagonal phases are implemented in software without physical pulses.Source: SINGLE QUTRIT GATE DECOMPOSITION · p. 7
GENERATION OF THE SINGLE-QUTRIT CLIFFORD GROUP
The single-qutrit Clifford group is generated from H and S and enumerated by a brute-force search up to its known size. The supplied discussion also notes a limitation for interleaving non-Clifford π/2 rotations.
GENERATION OF THE SINGLE-QUTRIT CLIFFORD GROUP
- A brute-force search generates the single-qutrit Clifford group from H and S, stopping at its known size of 216 elements.Generated matrices are added only if not already present up to a global phase; the search is described as impracticable for the two-qutrit Clifford group.Source: GENERATION OF THE SINGLE-QUTRIT… · p. 8
- π/2 rotations are not in the Clifford group, making their interleaved benchmarking less straightforward; characterizing non-Clifford gates is left for future work.The passage contrasts these rotations with π rotations and the Hadamard gate.Source: GENERATION OF THE SINGLE-QUTRIT… · p. 8
QUTRIT PAULI GROUP
The single-qutrit Pauli group is generated by Z and X, with Y and V added as defined operators. Each Pauli shares an eigenbasis with its Hermitian conjugate, permitting two expectation values per circuit.
QUTRIT PAULI GROUP
- The single-qutrit Pauli group is generated by Z and X, with Y = ZX and V = Z2X defined for convenience.For each group element, P2 = P† and P3 = I.Source: QUTRIT PAULI GROUP · p. 8
- Because each Pauli operator shares an eigenbasis with its Hermitian conjugate, two expectation values can be measured per circuit.The passage notes that each Pauli's square equals its Hermitian conjugate.Source: QUTRIT PAULI GROUP · p. 8
PAULI EXPECTATION VALUES MEASUREMENT
Pauli expectation values are obtained by rotating the computational basis into the operator’s eigenbasis, measuring populations, and combining outcomes as a weighted sum.
PAULI EXPECTATION VALUES MEASUREMENT
- For an N-qudit Pauli operator Q, B_Q rotates the computational basis into an eigenbasis of Q.Source: PAULI EXPECTATION VALUES MEASUREMENT · p. 8
- After applying B_Q†, the measurement protocol produces an outcome z in the eigenbasis.Source: PAULI EXPECTATION VALUES MEASUREMENT · p. 8
- The expectation value of Q is computed as a weighted sum over the measured populations.Source: PAULI EXPECTATION VALUES MEASUREMENT · p. 8