image/svg+xml Randomised Benchmarking
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image/svg+xml Project by Supervised by Dr Koh Teck Seng Zaw Lin Htoo Plus applications in the triple quantum dot qubit with Gate-Dependent Noise
image/svg+xml Noise in Quantum Computers
image/svg+xml The RB Protocol
image/svg+xml Triple Quantum Dot
image/svg+xml Noise in Quantum Computers
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image/svg+xml qubit
image/svg+xml state:
image/svg+xml |0⟩
image/svg+xml |0⟩
image/svg+xml |1⟩
image/svg+xml +
image/svg+xml −
image/svg+xml cos(θ/2)
image/svg+xml e sin(θ/2) iϕ
image/svg+xml |ψ⟩
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image/svg+xml X
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image/svg+xml representing data quantum states (“qubits”)
image/svg+xml logical operations unitary evolution (“gates”)
image/svg+xml ½–spin: |↑z⟩, |↓z⟩
image/svg+xml superpositionentanglement Quantum Phenomena
image/svg+xml should give
image/svg+xml Enormous Speedups database search (Grover, 1996)integer factoring (Shor, 1999)linear equations (Harrow et al., 2009)
image/svg+xml But quantum computers are inherently noisy! (Kalai 2011)
image/svg+xml Unable to carry out arbitrarily long & complex calculations
image/svg+xml Fault Tolerant Threshold Theorem
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image/svg+xml Example: Shor's Algorithm
image/svg+xml (Häner, Roetteler, & Svore, 2017)
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image/svg+xml m ~ 2,000 for a 2-bit number
image/svg+xml m ~ 8,000 for a 4-bit number
image/svg+xml Even with 0.1% error rate, accumulates to
image/svg+xml ~82% error rate for 2 bits
image/svg+xml ~99.97% error rate for 4 bits
image/svg+xml …
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image/svg+xml If error rate below certain threshold, arbitrarily longcalculations can be carried out (Aharonov & Ben-Or, 2008)
image/svg+xml “The entire content of the Threshold Theorem is thatyou're correcting errors faster than they're created.”
image/svg+xml — Scott Aaronson
image/svg+xml from Quantum Computing since Democritus (2013)
image/svg+xml Characterising Noise
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image/svg+xml Want to carry out a gate
image/svg+xml But actual implementedgate will have noise
image/svg+xml Noisy gate is ideal gatewith some noise process
image/svg+xml Goal is to characterise the errorrate of this noise process
image/svg+xml Quantum Process Tomography Treat noise process as a black box, measure how channeltransforms input to reconstruct process (Chuang & Nielsen, 1997) Chow, J. M., Gambetta, J. M., Tornberg, L., Koch, J., Bishop, L. S., Houck, A. A., ... Schoelkopf, R. J. (2009).Randomized Benchmarking and Process Tomography for Gate Errors in a Solid-State Qubit.Physical Review Letters, 102(9). https://doi.org/10.1103/physrevlett.102.090502 *(Chow et al., 2009) Chuang, I. L., & Nielsen, M. A. (1997). Prescription for experimental determination of thedynamics of a quantum black box. Journal of Modern Optics, 44(11-12), 2455–2467.https://doi.org/10.1080/09500349708231894
image/svg+xml Λ
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image/svg+xml Assumption:these statescan be preparedperfectly
image/svg+xml Assumption:these statescan be measuredperfectly
image/svg+xml Need to prepare and measure the state in every basis state
image/svg+xml Noise in the state preparation and measurement (SPAM)is included in Λ due to this assumption
image/svg+xml Due to SPAM errors, over-reports gate error (Chow et al., 2009),the key consideration for the fault-tolerant threshold theorem
image/svg+xml input ↑x ↑y ↑z ↓z
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image/svg+xml measure x, y, z
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image/svg+xml 2 −2 parameters 2n 4n
image/svg+xml not scalable
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image/svg+xml SPAM errors
image/svg+xml doesn't reflect gate error
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image/svg+xml How to characterise gate noise in a scalable mannerthat is independent of SPAM errors?
image/svg+xml Two main ideas
image/svg+xml 1. Accumulating Noise
image/svg+xml 2. Twirling
image/svg+xml Accumulating Noise
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image/svg+xml …
image/svg+xml Λ
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image/svg+xml Λ
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image/svg+xml Λ
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image/svg+xml As m increases, so does the noise, independently of SPAM
image/svg+xml But might accumulate differently
image/svg+xml for different types of noise
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image/svg+xml Possible to getconsistent behaviour?
image/svg+xml Twirling
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image/svg+xml “Average out” the noise channel
image/svg+xml = 1
image/svg+xml Example: Amplitude Damping
image/svg+xml Shrinks Bloch sphere towards ground state
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image/svg+xml p
image/svg+xml Shrinks Bloch sphere uniformly
image/svg+xml Depolarising channel
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image/svg+xml Same behaviour for all Λ
image/svg+xml Λ and have the same error rate
image/svg+xml (Magesan, Gambetta, & Emerson, 2012)
image/svg+xml The RB Protocol
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image/svg+xml … …
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image/svg+xml m times
image/svg+xml 1. Choose a random sequence of gates of length m
image/svg+xml 2. Define final inverting gate
image/svg+xml 3. Input a state, then measure the probability of obtaining the same state as Fm
image/svg+xml 4. Repeat with K random sequences of length m
image/svg+xml 5. Repeat with other values of m
image/svg+xml Each depolarising channel uniformly shrinks Blochsphere by a factor of p
image/svg+xml Survival probability F is fittedto an exponential model
image/svg+xml F(m) = p + m A B
image/svg+xml A B
image/svg+xml Coefficients A & B absorb SPAM errors
image/svg+xml r = (d−1)(1−p) d Average gate error
image/svg+xml Twirling — All noise types exhibit same behaviourFitted depolarising parameter recovers average gate error
image/svg+xml Depolarising Bitflip Random Unitary Λ Noise type Amplitude Damping
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image/svg+xml actualerror rate
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image/svg+xml SPAM Errors — Same gate error, different SPAM error Fitted depolarising parameter recovers average gate error
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image/svg+xml actualerror rate
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image/svg+xml Conclusion
image/svg+xml Advantages
image/svg+xml Scales polynomially with n
image/svg+xml (Magesan, Gambetta, & Emerson, 2012)
image/svg+xml A&B absorbs SPAM errors
image/svg+xml Disadvantages
image/svg+xml Only returns r
image/svg+xml Pathological cases(Proctor et al., 2017)
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image/svg+xml A likely candidate for large-scalequantum computation (Tarucha et al., 2016)
image/svg+xml — Long coherence times (Fujisawa et al., 2002)
image/svg+xml — Ability to initialise and measure spin (Ono, 2002)
image/svg+xml — Compatibility with mature semiconductor technology (Ono, Mori, & Moriyama 2019)
image/svg+xml The Triple Quantum Dot Qubit
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image/svg+xml Future Work
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image/svg+xml Conclusion
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image/svg+xml tl tr εm ε ε Adapted from Shim & Tahan (2016)
image/svg+xml States Spin of electronsin quantum dots
image/svg+xml Electrically controlledtunneling and detuning Logicaloperations
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image/svg+xml Introduced to control parameters
image/svg+xml Appears in quantum materials(Paladino et al., 2014)
image/svg+xml Ubiquitous in electronics(Wong, 2003)
image/svg+xml 1/f noise
image/svg+xml Sweet spot
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image/svg+xml Region where researchers havefound dephasing time to be longest(Russ & Burkard, 2017; Shim & Tahan, 2016)
image/svg+xml RB shows that operating in thesweet spot minimises error
image/svg+xml Sweet spot
image/svg+xml only detuning noise
image/svg+xml only tunneling noise
image/svg+xml Tunneling noise introduces fluctuations to the sweet spot behaviour
image/svg+xml Sensitivity to noise
image/svg+xml detuning noise
image/svg+xml tunneling noise
image/svg+xml Triple quantum dot qubit is more sensitive to tunneling noise
image/svg+xml attempts at inferring from the sweet spot behaviour
image/svg+xml ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 0.25 0.5 0.75 1
image/svg+xml attempts at interleavedbenchmarking (Magesan et al., 2012)
image/svg+xml Need other methods to differentiatedetuning and tunnelling noise
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image/svg+xml Improve pulse sequences
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image/svg+xml Existing error-correcting pulsesequences (Witzel & Sarma, 2007; Wang et al., 2014) not applicable
image/svg+xml Need coding scheme based onlimitations and noise characteristics
image/svg+xml Error-correcting pulse sequences,RB as a diagnostic tool (Zhang et al., 2017)
image/svg+xml Randomised benchmarking verifies sweet spot behaviour
image/svg+xml Reveals sensitivity of qubit to tunneling noise
image/svg+xml More work needs to be done todistinguish detuning and tunneling noise
image/svg+xml Diagnostic tool to develop and test pulse sequences
image/svg+xml References
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image/svg+xml Aaronson, S. (2013). Quantum Computing since Democritus. Cambridge University Press. Fujisawa, T., Austing, D. G., Tokura, Y., Hirayama, Y., & Tarucha, S. (2002). Allowed and forbidden transitions in artificial hydrogen and helium atoms.Nature, 419(6904), 278–281. https://doi.org/10.1038/nature00976 Häner, T., Roetteler, M., & Svore, K. M. (2017). Factoring using 2n+2 qubits with Toffoli based modular multiplication. Quantum Information andComputation, 17(7 & 8), 0673-0684. Kalai, G. (2011). How quantum computers fail: quantum codes, correlations in physical systems, and noise accumulation. arXiv preprint arXiv:1106.0485.Retrieved from https://arxiv.org/pdf/1106.0485 Grover, L. K. (1996). A fast quantum mechanical algorithm for database search. Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory ofComputing - STOC 96. https://doi.org/10.1145/237814.237866 Harrow, A. W., Hassidim, A., & Lloyd, S. (2009). Quantum Algorithm for Linear Systems of Equations. Physical Review Letters, 103(15).https://doi.org/10.1103/physrevlett.103.150502 Aharonov, D., & Ben-Or, M. (2008). Fault-tolerant quantum computation with constant error. SIAM Journal on Computing, 38(4), 1207–1282.https://doi.org/10.1137/s0097539799359385 Chow, J. M., Gambetta, J. M., Tornberg, L., Koch, J., Bishop, L. S., Houck, A. A., ... Schoelkopf, R. J. (2009). Randomized Benchmarking and ProcessTomography for Gate Errors in a Solid-State Qubit. Physical Review Letters, 102(9). https://doi.org/10.1103/physrevlett.102.090502 Chuang, I. L., & Nielsen, M. A. (1997). Prescription for experimental determination of the dynamics of a quantum black box. Journal of Modern Optics,44(11-12), 2455–2467. https://doi.org/10.1080/09500349708231894
image/svg+xml Tarucha, S., Yamamoto, M., Oiwa, A., Choi, B.-S., & Tokura, Y. (2016). Spin Qubits with Semiconductor Quantum Dots. Principles and Methods of QuantumInformation Technologies Lecture Notes in Physics, 541–567. https://doi.org/10.1007/978-4-431-55756-2_25 Shim, Y.-P., & Tahan, C. (2016). Charge-noise-insensitive gate operations for always-on, exchange-only qubits. Physical Review B, 93(12).https://doi.org/10.1103/physrevb.93.121410 Russ, M., & Burkard, G. (2017). Three-electron spin qubits. Journal of Physics: Condensed Matter, 29(39), 393001. https://doi.org/10.1088/1361-648x/aa761f Witzel, W. M., & Sarma, S. D. (2007). Multiple-Pulse Coherence Enhancement of Solid State Spin Qubits. Physical Review Letters, 98(7).https://doi.org/10.1103/physrevlett.98.077601 Wang, X., Bishop, L. S., Barnes, E., Kestner, J. P., & Sarma, S. D. (2014). Robust quantum gates for singlet-triplet spin qubits using composite pulses.Physical Review A, 89(2). https://doi.org/10.1103/physreva.89.022310 Zhang, C., Throckmorton, R. E., Yang, X.-C., Wang, X., Barnes, E., & Sarma, S. D. (2017). Randomized Benchmarking of Barrier versus Tilt Control of aSinglet-Triplet Qubit. Physical Review Letters, 118(21). https://doi.org/10.1103/physrevlett.118.216802 Proctor, T., Rudinger, K., Young, K., Sarovar, M., & Blume-Kohout, R. (2017). What Randomized Benchmarking Actually Measures. Physical Review Letters,119(13). https://doi.org/10.1103/physrevlett.119.130502 Shor, P. W. (1999). Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. SIAM Review, 41(2), 303–332.https://doi.org/10.1137/s0036144598347011 Paladino, E., Galperin, Y. M., Falci, G., & Altshuler, B. L. (2014). 1/f noise: Implications for solid-state quantum information. Reviews of Modern Physics,86(2), 361–418. https://doi.org/10.1103/revmodphys.86.361 Ono, K., Mori, T., & Moriyama, S. (2019). High-temperature operation of a silicon qubit. Scientific Reports, 9(1). https://doi.org/10.1038/s41598-018-36476-z Ono, K. (2002). Current Rectification by Pauli Exclusion in a Weakly Coupled Double Quantum Dot System. Science, 297(5585), 1313-1317.https://doi.org/10.1126/science.1070958 Magesan, E., Gambetta, J. M., & Emerson, J. (2012). Characterizing quantum gates via randomized benchmarking. Physical Review A, 85(4).https://doi.org/10.1103/physreva.85.042311
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