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Lectures on Fault-Tolerant Quantum Computation
B.M. Terhal, IBM Research
I. Descriptions of Noise and Quantum States
II. Quantum Coding and Error-Correction
III. Fault-Tolerant Error-Correction. Surface Codes. Some Results on Noise Thresholds.
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Some Background Reading
Textbook by Michael Nielsen & Isaac Chuang Lectures notes of John Preskill
PhD Thesis of Dan GottesmanPhD Thesis by Ben Reichardt
Arxiv: quant-ph/0110143 by Dennis, Landahl, Kitaev,Preskillon use of surface codesArxiv: quant-ph/0610063 by Aliferis, Cross on Bacon-Shor CodesArxiv: 0711.1556 by Cross, DiVincenzo, Terhal with threshold studies
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The 5 DiVincenzo Criteria1. Architecture needs to be scalable with well-defined
qubits.
2. Ability to initialize qubits to |00…0> state
3. Qubits should undergo little decoherence
4. Ability to enact a discrete set of logical gates. For example:
2-qubit gate: CNOT (C-X) or CPHASE (C-Z)1-qubit gates: Pauli X, Z, Y, Hadamard, Phase gate, T gate
5. Ability to measure single qubits in the computational (0,1) basis.
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Some Quantum Formalism
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Superoperators
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Examples
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Superoperator Noise Model
Each location in a quantum circuit is represented by its own superoperator which ideally is close to the idealoperation.
(Simplest) Noise Model considered in Fault-Tolerance Theory
In what situations is this model sufficient….
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(In)Sufficiency of Superoperator Noise Picture
Cross-talk between neighboring qubits (addressing the wrong qubits with the control fields)Couplings we cannot turn off…
Some noise is best modeled as a system interacting with a quantum environment. Correlations in time and space, non-markovian environment.
Some noise can be clearly approximated by classical fluctuations of control parameters, correlations in time and space of these parameters.
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Error Rates
Error rate of a superoperator?
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Error-CorrectionClassical error-correction is fairly common:
Satellite communication & deep-space communicationSoft (radiation errors) in dynamical RAM in satellitesCompact discs (Reed Solomon codes) and hard discs.Hard-wired coding of bits as ferromagnetic domains ->2D repetition code
Noise levels of quantum operations are high
Quantum error-correction will be crucial in any robust implementation of quantum computation.
Passive EC (e.g. topological quantum computation) or active
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Error-CorrectionClassical error-correction is fairly common:
Satellite communication & deep-space communicationSoft (radiation errors) in dynamical RAM in satellitesCompact discs (Reed Solomon codes) and hard discs.Hard-wired coding of bits as ferromagnetic domains ->2D repetition code
Noise levels of quantum operations are high
Quantum error-correction will be crucial in any robust implementation of quantum computation.
Passive EC (e.g. topological quantum computation) or active
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Classical Repetition Code
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Classical Repetition Code
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Quantizing Repetition Code
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Error Correction
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[[9,1,3]] Shor code
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Other Quantum Codes
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Other Quantum Codes
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Bacon-Shor CodesLattice of 3 x 3 qubits (or 5 x 5 or 7 x 7 etc.)X-stabilizers are 2 adjacent horizontal lines of Xs.Z-stabilizers (linear combinations of [[9,1,3]] stabilizers) are adjacent vertical lines of Zs.9-4=5 encoded qubits…(in general n2-2(n-1) qubits).
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Bacon-Shor CodesElements in N(S)-S of weight 2, no protection against thoseerrors...?
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Bacon-Shor CodesElements in N(S)-S of weight 2, no protection against thoseerrors...?
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Measuring Stabilizers
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Surface Code Family
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Surface Code Family
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Errors on Surface Codes
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Passive Noise Protection
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Noise Threshold
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Threshold Studies
Cross, DiVincenzo, Terhal, 2007
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What we study
Extended-rectangle.Here 1-Ga taken as transversal CNOT. Steane EC.Generate random X,Y,Z errors with probability p on all locations.Follow errors through gates in the rectangle. If errors add up to logical error between state at t and state at t’, call it a failure.
Estimate probability for failure p1 as function of p. Threshold: p1=p.
t’t
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Perfect Ancillas for Steane EC
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Thresholds
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Surface Code
We find for L , pc goes to 3.5 x 10-3
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Overhead versus logical error-rate