Quantum Circuits for Exact Unitary 
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:mi>t</mml:mi></mml:math>
-Designs and Applications to Higher-Order Randomized Benchmarking

Nakata, Yoshifumi; Zhao, Da; Okuda, Takayuki; Bannaĭ, Eiichi; Suzuki, Yôiti; Tamiya, Shiro; Heya, Kentaro; Yan, Zhiguang; Zuo, Kun; Tamate, Shuhei; Tabuchi, Yoshihiko; Nakamura, Y.

doi:10.1103/prxquantum.2.030339

Cited by 28 publications

(20 citation statements)

References 82 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Very efficient random circuit constructions for exact n -qubit 2-designs, where the resources required scale almost linearly with n , have also been devised 13 . More recently, random circuit constructions for general exact n -qubit t -designs were proposed 14 . However, they are only feasible for small systems, since the number of gates required scales exponentially with n and t for large n .…”

Section: Introductionmentioning

confidence: 99%

Implementation of single-qubit measurement-based t-designs using IBM processors

Strydom

Tame

2022

Sci Rep

View full text Add to dashboard Cite

Random unitary matrices sampled from the uniform Haar ensemble have a number of important applications both in cryptography and in the simulation of a variety of fundamental physical systems. Since the Haar ensemble is very expensive to sample, pseudorandom ensembles in the form of t-designs are frequently used as an efficient substitute, and are sufficient for most applications. We investigate t-designs generated using a measurement-based approach on superconducting quantum computers. In particular, we implemented an exact single-qubit 3-design on IBM quantum processors by performing measurements on a 6-qubit graph state. By analysing channel tomography results, we were able to show that the ensemble of unitaries realised was a 1-design, but not a 2-design or a 3-design under the test conditions set, which we show to be a result of depolarising noise during the measurement-based process. We obtained improved results for the 2-design test by implementing an approximate 2-design, in which measurements were performed on a smaller 5-qubit graph state, but the test still did not pass for all states. This suggests that the practical realisation of measurement-based t-designs on superconducting quantum computers will require further work on the reduction of depolarising noise in these devices.

show abstract

Section: Introductionmentioning

confidence: 99%

Implementation of single-qubit measurement-based t-designs using IBM processors

Strydom

Tame

2022

Sci Rep

View full text Add to dashboard Cite

show abstract

“…For quantum states, the other very common similarity measure is quantum fidelity, which induces distance between states known as Bures distance [23,24]. When one wants to compare unitary channel (quantum gate) with a general channel (noisy implementation of a gate), the relevant notions are worst-case [2] and average-case gate fidelity [25][26][27][28][29]. In both cases, the relevant optimization/averaging is over all quantum states, and the fidelity is a standard state For distance measures between measurements, one of the natural choices is to treat measurement as a quantum-classical channel and compute diamond norm distance [18,30].…”

Section: Related Workmentioning

confidence: 99%

“…First important question one can ask is how to estimate average-case quantum distances in easy-to-implement setting. Natural candidate seems to be randomized-benchmarking types of experiments, as they also employ unitary designs [29,41]. It would be also very interesting to connect quantum average-case distances with commonly used figures of merit used to assess quality of quantum devices.…”

Section: Open Problemsmentioning

confidence: 99%

“…It would be also very interesting to connect quantum average-case distances with commonly used figures of merit used to assess quality of quantum devices. Those include measures such as average fidelity [41] (which is perhaps the most widely used quality measure), or unitarity of quantum channels [29,73,74]. Furthermore, one can ask whether similar results can be obtained for different functions of output probability distributions such us classical fidelity or f -divergences [75,76].…”

Section: Open Problemsmentioning

confidence: 99%

See 1 more Smart Citation

Exploring Quantum Average-Case Distances: proofs, properties, and examples

Maciejewski¹,

Puchała²,

Oszmaniec³

2021

Preprint

View full text Add to dashboard Cite

In this work, we perform an in-depth study of recently introduced average-case quantum distances . The average-case distances approximate, up to relative error, the average Total-Variation (TV) distance between measurement outputs of two quantum processes, in which quantum objects of interest (states, measurements, or channels) are intertwined with random quantum circuits. Contrary to conventional distances, such as trace distance or diamond norm, they quantify average-case statistical distinguishability via random quantum circuits.We prove that once a family of random circuits forms an δ-approximate 4-design, with δ = o(d −8 ), then the average-case distances can be approximated by simple explicit functions that can be expressed via simple degree two polynomials in objects of interest. For systems of moderate dimension, they can be easily explicitly computed -no optimization is needed as opposed to diamond norm distance between channels or operational distance between measurements. We prove that those functions, which we call quantum average-case distances, have a plethora of desirable properties, such as subadditivity w.r.t. tensor products, joint convexity, and (restricted) data-processing inequalities. Notably, all of the distances utilize the Hilbert-Schmidt (HS) norm in some way. This gives the HS norm an operational interpretation that it did not possess before. We also provide upper bounds on the maximal ratio between worst-case and average-case distances, and for each of them we provide an example that saturates the bound. Specifically, we show that for each dimension d this ratio is at most d 1 2 , d, d 3 2 for states, measurements, and channels, respectively. To support the practical usefulness of our findings, we study multiple examples in which average-case quantum distances can be calculated analytically. Furthermore, we perform an extensive numerical study of systems up to 10 qubits where circuits have a form of variational ansätze with randomly chosen parameters. Despite the fact that such circuits are not known to form unitary designs, our results suggest that average-case distances are more suitable than the conventional distances for studying the performance of NISQ devices in such settings.

show abstract

“…The act of randomisation by the Haar-measure of the m-qubit unitary group is an important operation in quantum computation and information. It has found applications in, for instance, randomized benchmarking [1][2][3][4][5][6][7][8][9][10][11]50], quantum control, quantum data hiding [12], and channel twirling [50], to name a few. Besides practical applications, it plays a significant role in decoupling theorems, which is of immense significance in quantum information theory.…”

Section: Introductionmentioning

confidence: 99%

Approximate 3-designs and partial decomposition of the Clifford group representation using transvections

Singal¹,

Hsieh²

2021

Preprint

View full text Add to dashboard Cite

We study a scheme to implement an asymptotic unitary 3-design. The scheme implements a random Pauli once followed by the implementation of a random transvection Clifford by using state twirling. Thus the scheme is implemented in the form of a quantum channel. We show that when this scheme is implemented k times, then, in the k → ∞ limit, the overall scheme implements a unitary 3-design. This is proved by studying the eigendecomposition of the scheme: the +1 eigenspace of the scheme coincides with that of an exact unitary 3-design, and the remaining eigenvalues are bounded by a constant. Using this we prove that the scheme has to be implemented approximately O(m + log 1/ǫ) times to obtain an ǫ-approximate unitary 3-design, where m is the number of qubits, and ǫ is the diamond-norm distance of the exact unitary 3-design. Also, the scheme implements an asymptotic unitary 2-design with the following convergence rate: it has to be 1 sampled O(log 1/ǫ) times to be an ǫ-approximate unitary 2-design. Since transvection Cliffords are a conjugacy class of the Clifford group, the eigenspaces of the scheme's quantum channel coincide with the irreducible invariant subspaces of the adjoint representation of the Clifford group. Some of the subrepresentations we obtain are the same as were obtained in [36], whereas the remaining are new invariant subspaces. Thus we obtain a partial decomposition of the adjoint representation for 3 copies for the Clifford group. Thus, aside from providing a scheme for the implementation of unitary 3-design, this work is of interest for studying representation theory of the Clifford group, and the potential applications of this topic. The paper ends with open questions regarding the scheme and representation theory of the Clifford group.

show abstract

Quantum Circuits for Exact Unitary t -Designs and Applications to Higher-Order Randomized Benchmarking

Cited by 28 publications

References 82 publications

Implementation of single-qubit measurement-based t-designs using IBM processors

Implementation of single-qubit measurement-based t-designs using IBM processors

Exploring Quantum Average-Case Distances: proofs, properties, and examples

Approximate 3-designs and partial decomposition of the Clifford group representation using transvections

Contact Info

Product

Resources

About