Quantum Information Processing �with Finite Thermodynamic Resources

Contributed talk at Quantum Resources � Jake Xuereb, Dec 2023, Singapore.

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For the sake of introduction…

I’m a PhD student with the QUantum Information & Thermodynamics (QUIT Physics) Group

Qalypso Summer Schools�www.qalypso.info ��Qalypso 2024 : Precision in Quantum Information & Computation, Sep 2024

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“Information is not an abstract entity but exists only through a physical representation, thus tying it to all the restrictions and possibilities of our real physical universe.”��Rolf Landauer

Information is a physical entity, Physica A: Statistical Mechanics and its Applications 263, 63 (1999)

Some Motivation

How does thermodynamics restrict our ability to process quantum information?

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Something is resourceful is it allows us to accomplish a task well.�� Fidelity���Obtaining or generating a resource comes at a physical cost. �� Entropy Production

Setting

How does thermodynamics restrict our ability to process quantum information?

Talk Structure

Illustration above by Jeffrey Phillips for Cosmos Magazine while the rest are by the author.

Part 1 : Timekeeping as a resource in accurate quantum computation��Phys. Rev. Lett. 131,160204

Part 2 : Thermodynamic resources for measurement and decoding in Schumacher compression��arXiv:2311.14561

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1.1 - What’s a Clock really?

Erker, P.; Mitchison, M. T.; Silva, R.; Woods, M. P.; Brunner, N. et al. [..] (2017) Autonomous Quantum Clocks: Does Thermodynamics Limit Our Ability to Measure Time?. Physical Review X, Bd. 7 (3), S. 031022.

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1.1 - What’s a Clock really?

Erker, P.; Mitchison, M. T.; Silva, R.; Woods, M. P.; Brunner, N. et al. [..] (2017) Autonomous Quantum Clocks: Does Thermodynamics Limit Our Ability to Measure Time?. Physical Review X, Bd. 7 (3), S. 031022.

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1.2 - Three Perspectives

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1.3 - Tick Distributions & Quantum Control

The impact of imperfect timekeeping on quantum control, Xuereb et al, Phys. Rev. Lett. 131,160204 (2023).

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1.4 - Imperfect Timekeeping is Dephasing

Dephasing

Time-Evolution

study the impact of this channel…

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1.5 - Average Gate Fidelity

J. Emerson, R. Alicki, and K. Życzkowski, Scalable noise estimation with random unitary operators, Journal of Optics B: Quantum and Semiclassical Optics 7, S347 (2005)��M. A. Nielsen, A simple formula for the average gate fidelity of a quantum dynamical operation,�Physics Letters A 303, 249 (2002)

How much is imperfect timekeeping disturbing our computation?

Haar-Averaged Gate Fidelity of V given a noisy channel T

Nielsen’s Nice Formula

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1.6 - Clock Accuracy bounds the Fidelity

The impact of imperfect timekeeping on quantum control, Xuereb et al, Phys. Rev. Lett. 131,160204 (2023).

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1.5 - Clock Accuracy bounds the Fidelity

The impact of imperfect timekeeping on quantum control, Xuereb et al, �Phys. Rev. Lett. 131,160204 (2023).

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Bounding the average gate fidelity of composite channels using the unitarity�A. Carignan-Dugas, J. J. Wallman, and J. Emerson, New Journal of Physics 21, 053016 (2019).

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Proof Ideas : ��- i.i.d. Clocks �- Identical local dephasing channels�- Concatenation of non-destructive channels.�- Bound fidelity using unitarity (2 - norm)�

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- Paulis are involutary! So we find the unitarity exactly.

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The impact of imperfect timekeeping on quantum control, Xuereb et al, Phys. Rev. Lett. 131,160204 (2023).��Erker et. al Autonomous Quantum Clocks: Does Thermodynamics Limit Our Ability to Measure Time?. Physical Review X, Bd. 7 (3), S. 031022. (2017)

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Connecting this expression to Clock Accuracy in general is an open problem.��A model independent relationship between clock accuracy and entropy production is unknown.

1.5 - Clock Accuracy bounds the Fidelity

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arXiv:2311.14561

• Quantum Coding, Benjamin Schumacher, Phys. Rev. A 51, 2738 (1995).�
• Ideal Projective Measurements Have Infinite Resource Costs, Guryanova et. al, Quantum 4, 222 (2020).�
• The impact of imperfect timekeeping on quantum control, Xuereb et al, Phys. Rev. Lett. 131,160204 (2023).

Part 2 : Thermodynamics & Quantum Compression

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2.0 - Quantum Coding

• Quantum Coding, Benjamin Schumacher, Phys. Rev. A 51, 2738 (1995).

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2.1 - Quantum Coding with Finite Thermodynamic Resources

arXiv:2311.14561

• Quantum Coding, Benjamin Schumacher, Phys. Rev. A 51, 2738 (1995).�
• Ideal Projective Measurements Have Infinite Resource Costs, Guryanova et. al, Quantum 4, 222 (2020).�
• The impact of imperfect timekeeping on quantum control, Xuereb et al, Phys. Rev. Lett. 131,160204 (2023).

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2.2 - von Neumann Measurement Scheme

Ideal Projective Measurements Have Infinite Resource Costs, Guryanova et. al, Quantum 4, 222 (2020).��W. H. Zurek, Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?, Physical Review D 24, 1516 (1981)

Thinking of measurement as correlating with a probe…

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2.2 - von Neumann Measurement Scheme

Ideal Projective Measurements Have Infinite Resource Costs, Guryanova et. al, Quantum 4, 222 (2020).

Quantifies the correlations formed with the probe.��e.g.

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2.3 - Quantum Coding with Finite Thermodynamic Resources

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• Quantum Coding with Finite Thermodynamic Resources, Xuereb et al. arXiv:2311.14561 (2023)

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• An improved Landauer principle with finite-size corrections, Reeb & Wolf, New J. Phys. 16 103011 (2014).�
• Energy-Consumption Advantage of Quantum Computation, Meier & Yamasaki, arXiv:2305.11212 (2023).�
• Quantum Coding with Finite Thermodynamic Resources, Xuereb et al. arXiv:2311.14561 (2023)

2.4 - A Cooling Protocol to improve our resources

Go To Flo’s Talk on Friday!

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• An improved Landauer principle with finite-size corrections, Reeb & Wolf, New J. Phys. 16 103011 (2014).�
• Energy-Consumption Advantage of Quantum Computation, Meier & Yamasaki, arXiv:2305.11212 (2023).�
• Quantum Coding with Finite Thermodynamic Resources, Xuereb et al. arXiv:2311.14561 (2023)

2.4 - A Cooling Protocol to improve our resources

Decoding with thermal qubits

Measuring with thermal qubit probes

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Summary

Illustration above by Jeffrey Phillips for Cosmos Magazine while the rest are by the author.

1. Listen to Landauer, the free operations of your resource theory are not really free.�
2. Achieving better timekeeping means more entropy production or slower gates.

Better timekeeping, higher gate complexity.�

1. You can improve your states by cooling them at cost of entropy production.�
2. Accurate QIP comes at the cost of entropy production.

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Thanks to my collaborators and mentors!

Pauli Erker

Marcus Huber

Florian �Meier

Mark �Mitchison

Steve Campbell

John Goold

André Xuereb

Tiago Debarba

Jake Xuereb

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Thank you very much

for your kind attention.��Questions?

jake.xuereb.@tuwien.ac.at

www.jakexuereb.com

@curlyqubit

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4.2 - Fidelity Scaling Relationships

For a fidelity threshold of 0.5…��1,000,000 CNOTs N = 1,000,000�10,000 CNOTs N = 25,000��Experimentally, think of a pulse with τ=100ns�we’d achieve these accuracies with time uncertainties ��σ = 0.1 ns N = 1,000,000�σ = 0.633 ns N = 25,000��Electronic jitter in current timekeeping control systems (e.g. ARTIQ Sinara) is 0.3 ns ….��But what is σ really? How do we measure it?�

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4.3 - Quantum Coding with Finite Thermodynamic Resources

arXiv:2311.14561

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2.0 - von Neumann Measurement Scheme

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4.3 - Cooling a Qubit with an Imperfect Clock

J. Xuereb, F. Meier, P. Erker, M. T. Mitchison and M. Huber, “The impact of imperfect timekeeping on quantum control” arXiv:2301.1076 (2023)��F. Clivaz, R. Silva, G. Haack, J. B. Brask, N. Brunner, and M. Huber, Unifying paradigms of quantum refrigeration: Fundamental limits of cooling and associated work costs, Physical Review E 100 (2019)��R. Silva, G. Manzano, P. Skrzypczyk, and N. Brunner,Performance of autonomous quantum thermal machines:Hilbert space dimension as a thermodynamical resource, Physical Review E 94 (2016)

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3.4 - The Result for a single gate

• Clock Accuracy exponentially impacts the achievable average gate fidelity.
• This is dependent on the size of your path on an effective Bloch Sphere.
• For the same uncertainty, trying to do a gate faster is going to give you lower fidelity.

J. Xuereb, F. Meier, P. Erker, M. T. Mitchison and M. Huber, “The impact of imperfect timekeeping on quantum control” arXiv:2301.1076 (2023)��F. Meier, E. Schwarzhans, P. Erker, and M. Huber,“Fundamental accuracy-resolution trade-off for time-

keeping devices,” Phys. Rev. Lett. 131, 220201 (2023)��V. M. Schäfer, C. J. Ballance, K. Thirumalai, L. J.Stephenson, T. G. Ballance, A. M. Steane, and D. M.

Lucas, “Fast quantum logic gates with trapped-ion qubits” Nature 555, 75–78 (2018).

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3.4 - Average Gate Fidelity

J. Emerson, R. Alicki, and K. Życzkowski, Scalable noise estimation with random unitary operators, Journal of Optics B: Quantum and Semiclassical Optics 7, S347 (2005)��M. A. Nielsen, A simple formula for the average gate fidelity of a quantum dynamical operation,�Physics Letters A 303, 249 (2002)

How much is imperfect timekeeping disturbing our computation?

State Fidelity

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3.4 - Average Gate Fidelity

J. Emerson, R. Alicki, and K. Życzkowski, Scalable noise estimation with random unitary operators, Journal of Optics B: Quantum and Semiclassical Optics 7, S347 (2005)��M. A. Nielsen, A simple formula for the average gate fidelity of a quantum dynamical operation,�Physics Letters A 303, 249 (2002)

How much is imperfect timekeeping disturbing our computation?

Gate Fidelity of V given a noisy channel T