1. A Universal Operator Theoretic Framework for Quantum Fault Tolerance Yaakov S. Weinstein MITRE Quantum Information Science Group MITRE Quantum Error Correction 2007 University of Southern California Approved for Public Release; Distribution Unlimited: 07-1576; © 2007 The MITRE Corporation

2. Why Operator Quantum Fault Tolerance? Similar to the way operator quantum error correction (OQEC) is a universal operator framework for quantum error correction Provides top-down approach to fault tolerance starting with the complete dynamics of the system Establishes operator theoretic foundation connecting quantum error correction and quantum fault tolerance Improved accuracy of error threshold values Operator Quantum Fault Tolerance (OQFT) is a universal operator framework for quantum fault tolerance MITRE

3. OQEC: EAOQEC: Quantum Error Correction Hierarchy QEC: MITRE Initial state: State to protect ancilla Liouvillian Superoperators Linear QEC: P Complete system dynamics due to error and recovery P  -  = 0 Error recovery condition:

4. From QEC to Quantum Fault Tolerance P  -  = 0(Linear QEC) Experimental quantum computation will never work perfectly (though it may work fault tolerantly)… Let’s quantify how far from perfect the implementation is and bound it by a prescribed accuracy… QEC protects quantum information as it traverses a channel. Quantum Fault Tolerance (QFT) applies to the general case of quantum computation, involving some (unitary) dynamics, U Transform from Liouville space to ‘Kraus-like’ Operator space: (Linear QEC) For positive maps E k is not in general equal to E' k Where {A k, A' k } ≠ {UA k, UA' k } are Kraus-like operators for open system dynamics intended to implement U MITRE

5. The Quantum Computer Condition (QCC) The implementation inaccuracy serves as a metric quantifying the success of applied error correction and avoidance methods The QCC includes all types of quantum error correction and avoidance as special cases when: U = I,  = 0 The QCC forms the basis of our “top-down” approach to operator fault tolerance based on system level dynamical constraints full specification of actual system dynamics MITRE desired dynamics Implementation Inaccuracy Bound on implementation inaccuracy (prescribed) Schatten-1 norm

6. Operator Quantum Fault Tolerance Top-down approach to fault tolerance based on system-level dynamical constraints full specification of system dynamics with prescribed success criterion,  characterization of system dynamics at i th level of concatenation (may not be the same at each level) MITRE Operator Quantum Fault Tolerance (OQFT)

7. Example: Local Stochastic Noise Implementation inaccuracy at concatenation level i becomes: OQFT success criterion FT success criterion OQFT provides fine-grained resolution for error thresholds MITRE System dynamics

8. More Than One Noise Generator Let’s look a bit more carefully… For simplicity, assume perfect syndrome measurements… Error model: any physical qubit will be rotated by  x with probability p x,  y with probability p y and  z with probability p z. The final state of a logical qubit will then be: Depends on the p k, choice of code, recovery procedure, etc. Implementation Inaccuracy: [5,1,3] code Three noise generators of different strengths for second level of concatenation MITRE pxpx pypy pzpz

9. Conclusion & Future Work The QCC provides a unifying picture for all forms of quantum error correction and avoidance. The approach presented provides a useful metric in quantifying the accuracy of a given quantum information science implementation The QCC provides a universal operator-theoretic framework for quantum fault tolerance based on a top-down criterion, providing flexibility relative to the noise model and leading to improved accuracy for error threshold values Continue exploring different error models, error correcting and avoidance codes, etc. Calculate actual error threshold values MITRE

10. Reliable Final Computational Results from Faulty Quantum Computation G. Gilbert, M. Hamrick, Y.S. Weinstein (MITRE), arXiv:0707.0008, submitted to NJP MITRE Funded by: MITRE Technology Program (GG, MH, YSW) National Science Foundation (VA, ARC) Related Papers A Universal Operator Theoretic Framework for Quantum Fault Tolerance G. Gilbert, M. Hamrick, Y.S. Weinstein (MITRE), V. Aggarwal, A. Robert Calderbank (Princeton), arXiv:0709.0128, submitted to PRA

11. Example MITRE END

12. Quantum Computational Path

13. MITRE T(H logical ) G F p OmOm ImIm XY ideal (unitary) quantum computation: Accuracy of Quantum Computational Results (1) But real quantum computers don’t implement U exactly. What’s the effect on the failure probability bound of implementation errors? Probability that the ideal quantum computation followed by measurement produces outcome y: Probability that the ideal quantum computation followed by measurement produces the correct result F(x): The quantity p bounds the probability that the ideal quantum computation followed by measurement fails to produce the correct result Kitaev’s Model of Ideal Quantum Computation Kitaev (1997)

14. MITRE T(H comp ) P G  T(H logical ) M {l→c} M {c→l} Accuracy of Quantum Computational Results (2) Kitaev model gives failure probability for ideal computation. QCC describes deviation of the quantum computation from the ideal. What’s the combined effect? The Quantum Computer Condition (QCC) Gilbert, Hamrick & Weinstein (2007) Implementation Inaccuracy Ideal Computation Actual Computation Non-dynamical maps linking the logical and computational Hilbert spaces Notational Simplification:

15. MITRE T(H logical ) G F p OmOm ImIm XY T(H comp ) P G  T(H logical ) M {l→c} M {c→l} T(H comp ) P F Y X OmOm  + p M {c→l} M {l→c} ImIm Accuracy of Quantum Computational Results (3) Kitaev model shows ideal quantum computation can produce correct output, within p the QCC states that real quantum computation can realize ideal quantum computation, within  our result shows that real quantum computation can produce correct output, within p +  Composite diagram commutes with probability Failure probability

16. Achieving a Specified Probability of Success We can now derive a bound on the implementation inaccuracy that will guarantee the correct result with a specified probability –End user specifies maximum probability of failure – means majority voting can be used –Our result bounds total probability of failure –Bounds p b on failure probability, p, for ideal quantum computation are known for algorithms of interest: –e.g., in Grover search for 1 out of n items –We require that the implementation error tolerance be bounded by –It follows that the probability that the computation fails to return the correct result is bounded, as required, by MITRE We wish to achieve But the standard theory of fault tolerance yields a bound on the probability that the quantum computation results in the correct quantum state, not the implementation inaccuracy, which is a bound on the error norm of the state. Can we relate the two?

17. Fault Tolerance Theory Fault tolerance techniques enable correct dynamical evolution of the quantum computation in the presence of noise –Circuit is subject to (random) errors –Techniques such as concatenated quantum error correcting codes are used to correct errors –Fault tolerant circuit operates on encoded states –Not all errors are corrected, only the more likely ones –There is a residual error probability that the circuit fails to produce the desired quantum state as its output –Residual error probability can be estimated from theory –Doesn’t address quantum uncertainties in the final measurement –Need to find connection between residual error probability and probability of obtaining the correct final result Residual Error Probability Elementary gate failure probability Error threshold (function of circuit structure) Number of levels of concatenation (for a single logical gate using simplified model for concatenated error codes) MITRE Preskill (1998)

18. Specifying Fault Tolerance to Guarantee Probability of Obtaining Correct Final Result Fault tolerance theory yields a residual error probability –N g is the number of logical gates in the formal description of the algorithm We derive from this a bound on implementation inaccuracy given by: Success criterion is achieved if this bound satisfies For a given circuit and a given probability of elementary gate failure, this gives the number of levels of concatenation required to achieve the success criterion for the overall result MITRE

19. Example MITRE Number of gates in circuit Failure probability tolerance for final result Measurement failure bound for quantum algorithm Fault tolerant error threshold Elementary Gate Failure Probability Levels of Concatenation Levels of Concatenation Required to Guarantee Specified Tolerance of 0.4 for Failure to Compute Correct Final Result