Performance Requirements of a Quantum Computer Using Surface Code Error Correction

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1 Performance Requirements of a Quantum Computer Using Surface Code Error Correction Cody Jones, Stanford University Rodney Van Meter, Austin Fowler, Peter McMahon, James Whitfield, Man-Hong Yung, Thaddeus Ladd, Alán Aspuru-Guzik, Jungsang Kim, Yoshihisa Yamamoto 2 nd International Conf. on Quantum Error Correction December 7, 2011, Los Angeles

2 Problem Statement Fully account for the resources in large-scale quantum computing Examine the overhead costs for all fault-tolerant preparation steps Determine the implications for hardware performance

3 Problem Statement

4 Layered Architecture for Quantum Computing

5 Layered Architecture for Quantum Computing

6 Layered Architecture for Quantum Computing

7 Layered Architecture for Quantum Computing

8 Layered Architecture for Quantum Computing

Physical Layer Physical Qubit Self-assembled InAs quantum dot Imamoglu, et al.

83, 4204 (1999) Laser pulses Physical Gate Stimulated Raman transition with

Nature 456 218-221 (2008) Measurement Dispersive optical spin measurement

9 Physical Layer Physical Qubit Self-assembled InAs quantum dot Imamoglu, et al. Phys. Rev. Lett. 83, 4204 (1999) Laser pulses Physical Gate Stimulated Raman transition with ultrafast broadband pulse Press, et al. Nature (2008) Measurement Dispersive optical spin measurement Atatüre, et al. Nature Physics 3, 101 (2007) Coherence time (T 2 ~ 3 μs) [Press, et al. Nature Photonics 4, (2010)] h Gate execution times ( = ps) gµ B B Errors systematic or random?

Virtual Layer Cause destructive interference of systematic errors Dynamical decoupling sequence to correct dephasing errors T 2 * is very fast, so embed

10 Virtual Layer Cause destructive interference of systematic errors Dynamical decoupling sequence to correct dephasing errors T 2 * is very fast, so embed DD at lowest level BB1 compensation sequence to correct gate errors, such as laser intensity fluctuations Wimperis, J. Magn. Reson. Ser. B 109, 221 (1994)

11 Virtual Layer

12 Quantum Error Correction Layer Surface code: estimate distance needed Extract Syndrome Fowler, et al. Phys. Rev. A 80, (2009) Syndrome Matching Estimating Surface Code Distance Fowler, et al. arxiv/ (2011) ε thresh ε V 10-3 C 0.03 ε L d 29

13 Quantum Error Correction Layer

14 Logical Layer Use fault-tolerant QEC to deliver any arbitrary gate to the Application Layer State Distillation Bravyi and Kitaev, Phys. Rev. A 71, (2005) Approximating Arbitrary Quantum Gates 15 faulty ancillas 1 purified ancilla Methods with and without ancillas

15 State Distillation Ancilla states required to make universal gate set Need high-fidelity for fault-tolerance (e.g ) Distillation Circuit Concatenation Ancilla is consumed by this circuit, so we need very many ancillas at logical infidelity ~ Quantum computers will require factories to produce these ancilla as needed

16 Resource Analysis for State Distillation Fidelity improvement: p(error) ~35p 3 e.g., 2 levels distillation: [p 0 = 10-3 ] [p 2 = ] Distillation Levels Min. Circuit Depth Circuit Volume Leadingorder Error 1 Level 2 Levels 3 Levels 6x CNOT 12x CNOT 18x CNOT 72 qubits gates 1152 qubits gates qubits gates 35p 3 (1.5E6) p 9 (1.2E20) p 27

17 Arbitrary Quantum Gates Use finite gate set from Layer 3 to approximate any arbitrary gate within precision ε Gate Sequences (no ancilla) Gate sequence methods approximate a desired gate with fundamental gates from Layer 3 Fowler, QIC 11, (2011) Phase Kickback (multi-qubit ancilla) Phase kickback uses a special ancilla state to perform phase gates Although requires more qubits, can have lower circuit depth Kitaev, Shen, and Vyalyi, Classical and Quantum Computation, AMS (2002)

18 Gate Sequence Methods Approximate desired gate U with some sequence of gates in fundamental set Longer sequences produce better approximations at the expense of circuit depth and more T gates 1 0 e 0 π i 8 Circuit Depth Solovay-Kitaev Fowler s Method Phase Kickback O(log c (1/ε)) 3 < c < 4 O(log(1/ε)) RC: O( log(1/ε) ) CL: O( log(log(1/ε)) ) Calculation Time O(poly(log(1/ε))) O(poly(1/ε)) O(1) [negligible]

19 Solovay-Kitaev is Expensive

20 Resource Analysis for Arbitrary Gates Solovay-Kitaev appears to never produce an advantageous sequence Fowler s method requires exhaustive search (dashed lines extrapolated)

21 Separation in Time Scales Operation times increase by orders of magnitude from Physical to Logical layer

22 Shor s Algorithm Assumptions Optical quantum dots Surface code QEC Shor implementation given in [Van Meter, et al. IJQI 8, 295 (2010)] ε V = 10-3 / ε thresh = Depth d = 35 Fixed size: 10 5 logical qubits Algorithm stalls when distillation is not fast enough Require ~90% of QC devoted to distillation

23 Quantum Simulation (First-Quantized) See poster by James Whitfield Assumptions Optical quantum dots Surface code QEC First-quantized simulation algorithm for energy eigenvalue given in [Kassal et al. PNAS 105, (2008)] ε V = 10-3 / ε thresh = Depth d = simulated time steps

24 Quantum Simulation (Second-Quantized) LiH energy eigenvalue using STO-3G basis Assumptions Optical quantum dots Surface code QEC Second-quantized simulation algorithm for energy eigenvalue given in [Whitfield et al. Molecular Physics 109, (2011)] ε V = 10-3 / ε thresh = Depth d = 31, 31, 45 (different traces) 1000 simulated time steps

25 Conclusions A layered architecture framework facilitates the design of fault-tolerant quantum computers The overhead costs associated with faulttolerance separate operation times at physical and logical layers by 4-6 orders of magnitude Physical gates must be fast (sub-microsecond) Further reading: Layered architecture for quantum computing [arxiv: ] Simulating chemistry efficiently on fault-tolerant quantum computers [in preparation]

26 Auxiliary Slides

27 Layered Architecture

28 Hadamard Pulses in Quantum Dots Laser pulse that causes X-axis precession in physical qubit at same rate as Z-axis precession from magnetic field By pairing two Hadamard pulses with a variable delay in between (Z rotation), we can create high-fidelity X rotations

29 8H Decoupling Sequence Dynamical decoupling sequence similar to CPMG, tailored to optical quantum dots Removes systematic errors to first-order in control and dephasing bath

30 S = exp(iπ/4 σ z ) Phase Gate without Measurement S-gate = i without measurement: Still requires an ancilla state (which must be injected and distilled) However, this ancilla can be re-used

31 Quantum Dot Architecture Experimental Apparatus

32 Phase Kickback (Kitaev-Shen-Vyalyi) Use multi-qubit ancilla for phase gate rotations This ancilla is an eigenstate of addition; the eigenvalue is a phase rotation: When controlled-addition is performed on the ancilla, a phase is kicked back to the control qubit:

33 Phase Kickback in Simulation

34 Phase Kickback w/ Carry-Lookahead

Overview of Topological Cluster-State Quantum Computation on 2D Cluster-State

Overview of Topological Cluster-State Quantum Computation on 2D Cluster-State based on High-threshold universal quantum computation on the surface code -Austin G. Fowler, Ashley M. Stephens, and Peter