Sampling Overhead Analysis of Quantum Error Mitigation: Uncoded vs. Coded Systems

Xiong, Yifeng; Chandra, Daryus; Ng, Soon Xin; Hanzo, Lajos

doi:10.1109/access.2020.3045016

Cited by 19 publications

(19 citation statements)

References 57 publications

(100 reference statements)

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“…The error reduction capability of QEM comes at the price of a computational overhead. To elaborate, QEM is implemented by sampling from a "quasi-probability representation" of the inverse channel, which would increase the variance of the computational results, hence some computational overhead (termed as "sampling overhead" [21], [22]) is required for ensuring that a satisfactory accuracy can be achieved. By appropriately choosing the total number of samples, one may strike a beneficial computational accuracy vs. overhead trade-off.…”

Section: Our Contributions ✓ Monte Carlo-based Channel Inversion Anal...mentioning

confidence: 99%

“…To summarize, these results imply that when the quantum circuit is long in depth or large in the number of qubits, the computational error become excessive in practical applications. [19] × No QEM Analytical and numerical S. Wang et al [20] ✓ No QEM Analytical and numerical S. Endo et al [21] ✓ Exact channel inversion Sampling overhead vs. accuracy trade-off Only numerical Y. Xiong et al [22] ✓ Exact channel inversion Analytical and numerical R. Takagi [23] ✓ Exact channel inversion Analytical and numerical…”

Section: Introductionmentioning

confidence: 99%

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The Accuracy vs. Sampling Overhead Trade-off in Quantum Error Mitigation Using Monte Carlo-Based Channel Inversion

Xiong

Hanzo

2022

IEEE Trans. Commun.

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Quantum error mitigation (QEM) is a class of promising techniques for reducing the computational error of variational quantum algorithms. In general, the computational error reduction comes at the cost of a sampling overhead due to the variance-boosting effect caused by the channel inversion operation, which ultimately limits the applicability of QEM. Existing sampling overhead analysis of QEM typically assumes exact channel inversion, which is unrealistic in practical scenarios. In this treatise, we consider a practical channel inversion strategy based on Monte Carlo sampling, which introduces additional computational error that in turn may be eliminated at the cost of an extra sampling overhead. In particular, we show that when the computational error is small compared to the dynamic range of the error-free results, it scales with the square root of the number of gates. By contrast, the error exhibits a linear scaling with the number of gates in the absence of QEM under the same assumptions. Hence, the error scaling of QEM remains to be preferable even without the extra sampling overhead. Our analytical results are accompanied by numerical examples. NOTATIONS• Deterministic scalars, vectors and matrices are represented by x, x, and X, respectively, whereas their random counterparts are denoted as x, x, and X, respectively. Deterministic sets, random sets, and operators are denoted as X , X, and X , respectively. • The notations 1 n , 0 n , 0 m×n , and I k , represent the ndimensional all-one vector, the n-dimensional all-zero vector, the m × n dimensional all-zero matrix, and the k × k identity matrix, respectively. The subscripts may be omitted if they are clear from the context. • The notation ∥x∥ p represents the ℓ p -norm of vector x, and the subscript may be omitted when p = 2. For matrices, ∥A∥ p denotes the matrix norm induced by the corresponding ℓ p vector norm. The notation 1/x represents the element-wise reciprocal of vector x. • The notation [A] i,j denotes the (i, j)-th entry of matrix A. For a vector x, [x] i denotes its i-th element. The submatrix obtained by extracting the i 1 -th to i 2 -th rows and the j 1 -th to j 2 -th columns from A is denoted as [A] i1:i2,j1:j2 . The notation [A] :,i represents the i-th column of A, and [A] i,: denotes the i-th row, respectively. • The notation A ⊗ B represents the Kronecker product between matrices A and B. The notation A ⊙ B denotes the Hadamard product between matrices A and B.

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Section: Our Contributions ✓ Monte Carlo-based Channel Inversion Anal...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Accuracy vs. Sampling Overhead Trade-off in Quantum Error Mitigation Using Monte Carlo-Based Channel Inversion

Xiong

Hanzo

2022

IEEE Trans. Commun.

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“…Here, to accommodate computability and expressibility at the same time, we take another approach considered in Ref. [33,66]: we choose P to be all physical quantum channels. We notice that the norm • provides the cost of error mitigation in this setting as γ CPTP (Θ) = Θ −1 = Θ −1 ♦ , which can be efficiently computed by semidefinite programming.…”

Section: Error Mitigationmentioning

confidence: 99%

Operational applications of the diamond norm and related measures in quantifying the non-physicality of quantum maps

Regula

Takagi

2021

Quantum

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Although quantum channels underlie the dynamics of quantum states, maps which are not physical channels — that is, not completely positive — can often be encountered in settings such as entanglement detection, non-Markovian quantum dynamics, or error mitigation. We introduce an operational approach to the quantitative study of the non-physicality of linear maps based on different ways to approximate a given linear map with quantum channels. Our first measure directly quantifies the cost of simulating a given map using physically implementable quantum channels, shifting the difficulty in simulating unphysical dynamics onto the task of simulating linear combinations of quantum states. Our second measure benchmarks the quantitative advantages that a non-completely-positive map can provide in discrimination-based quantum games. Notably, we show that for any trace-preserving map, the quantities both reduce to a fundamental distance measure: the diamond norm, thus endowing this norm with new operational meanings in the characterisation of linear maps. We discuss applications of our results to structural physical approximations of positive maps, quantification of non-Markovianity, and bounding the cost of error mitigation.

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“…Post-selection increases the difficulty of implementing basis operations as there needs to be a conditional statement-even before post-processing of Monte Carlo results-that accepts the measurement results when some condition is met, or reject otherwise. One could choose a more general class of basis operations that includes a continuous set of noisy implementable operations [81][82][83][84][85], which could reduce the sampling cost characterised in Eq. (6.12).…”

Section: Gate Set Tomographymentioning

confidence: 99%

“…In addition, different basis operations can be used for quantum error mitigation. Potential future work can be focused on exploring the use of a continuous set of noisy implementable operations [81][82][83][84][85] which may reduce the C-coefficient. Another potential outlook is to explore if error mitigation methods may be applied to overcome noise due to crosstalk [124].…”

Section: -Frank Herbertmentioning

confidence: 99%

Quantum inference protocol : from origination to application

Ho¹

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First and foremost, a disclaimer. There are many people I wish to thank but there is a word limit on this thesis. Yes, this includes the acknowledgements. It's a pity. I could have written a novel. Anyway, here goes.My deepest gratitude to my supervisor Asst. Prof. Mile Gu and co-supervisor Assoc. Prof. Chew Lock Yue.To Mile, your creativity and ability to simplify the most complex concepts has inspired me greatly. Thank you for graciously granting the countless opportunities for engagement with other colleagues in the world. You have definitely helped to broaden my perspectives by-without any hints of hesitation-sponsoring trips to conferences and workshops in Madrid, Korea, and Shenzhen when travel was possible.To Lock Yue, thank you for the inspiration and advice on research while walking with me, providing advice in life and in the PhD. Your passion for understanding the fundamentals of science has impacted me positively and I am thankful and grateful for your supervision.To Thomas, or more personally known as Tom, I count myself extremely lucky to have started the PhD journey under your mentorship. Thank you for guiding me in the first half of my PhD stint with patience and constructive criticism. I'm glad we stuck to weekly discussions from the beginning that saw us tearing down my arguments and shaping my thought processes. I have learnt so much. I'll never forget our trip to Korea for a conference and the many conversations about coffee, tea, cats, and life! To Jeremiah, a beloved brother whom I have known for the past two decades. Bro, I could not have asked for a better person to celebrate joyful times and go through rough times with. You've seen me through this PhD journey, listening to my gripes and struggles, giving life advice along the way. You always inspire me to be a better person every day. I can't believe we have known each other for two decades. I'm excited for what the future holds for our friendship as we venture forward onto different life stages. I won't forget our special bromantic moments we had vacuuming your car from the suspected roach infestation or enjoying the sunset with Kim and vii Kat back in mid-2020! To Huan, biggest appreciation for walking with me through fun times and providing counsel when the path was seemingly very narrow. I owe you a long-overdue catch up session over prata and tea! Special mention to a few people. Wei Liang, a big brother and fellow musician, bassist, and life advisor, thank you for the many great times and for proof-reading this entire thesis; Guo Yao and Minjeong, my colleagues and badminton buddies who keep me striving for excellence on the badminton court, I will definitely miss badminton with you guys; special thanks to Minjeong for meticulously scrutinising the details of this thesis as well; Grayson, my fellow cross-island travel buddy and PhD colleague that I rant to during our cross-island journeys, whom I've known since undergraduate days, travelling to Sweden with and eating burnt cooking countless times; Ryuji, for the inspiration and enth...

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Sampling Overhead Analysis of Quantum Error Mitigation: Uncoded vs. Coded Systems

Cited by 19 publications

References 57 publications

The Accuracy vs. Sampling Overhead Trade-off in Quantum Error Mitigation Using Monte Carlo-Based Channel Inversion

The Accuracy vs. Sampling Overhead Trade-off in Quantum Error Mitigation Using Monte Carlo-Based Channel Inversion

Operational applications of the diamond norm and related measures in quantifying the non-physicality of quantum maps

Quantum inference protocol : from origination to application

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