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
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
Problem Statement
Layered Architecture for Quantum Computing
Layered Architecture for Quantum Computing
Layered Architecture for Quantum Computing
Layered Architecture for Quantum Computing
Layered Architecture for Quantum Computing
Physical Layer Physical Qubit • Self-assembled InAs quantum dot • Imamoglu, et al. Phys. Rev. Lett. 83 , 4204 (1999) Laser pulses Measurement Physical Gate • Dispersive optical spin measurement • Stimulated Raman transition with ultrafast • Atatüre, et al. Nature Physics 3 , 101 (2007) broadband pulse • Press, et al. Nature 456 218-221 (2008) • Coherence time (T 2 ~ 3 μ s) [Press, et al. Nature Photonics 4 , 367-370 (2010)] h • Gate execution times ( 20 – 40 ps) = µ g B • Errors – systematic or random? B
Virtual Layer Cause destructive interference of systematic errors Dynamical decoupling sequence to correct dephasing errors * is very fast, so embed DD • T 2 at lowest level BB1 compensation sequence to correct gate errors, such as laser intensity fluctuations • Wimperis, J. Magn. Reson. Ser. B 109 , 221 (1994)
Virtual Layer
Quantum Error Correction Layer Surface code: estimate distance needed Extract Syndrome Syndrome Matching Fowler, et al. Phys. Rev. A 80 , 052312 (2009) Estimating Surface Code Distance Fowler, et al. ε thresh 9×10 -3 arXiv/1110.5133 ε V 10 -3 (2011) C 0.03 ε L 10 -15 d 29
Quantum Error Correction Layer
Logical Layer Use fault-tolerant QEC to deliver any arbitrary gate to the Application Layer Approximating Arbitrary State Distillation Quantum Gates • Bravyi and Kitaev, Phys. Rev. A 71 , 022316 (2005) 1 purified ancilla Methods with and 15 faulty without ancillas ancillas
State Distillation Ancilla states required to make universal gate set Need high-fidelity for fault-tolerance (e.g. 10 -15 ) Distillation Circuit Concatenation Ancilla is consumed by this circuit, so we need very many ancillas at logical infidelity ~ 10 -15 Quantum computers will require “factories” to produce these ancilla as needed
Resource Analysis for State Distillation Fidelity improvement: p (error) ~35 p 3 ◦ e.g., 2 levels distillation: [ p 0 = 10 -3 ] [ p 2 = 1.5 ×10 -21 ] Distillation 1 Level 2 Levels 3 Levels Levels Min. Circuit 6x CNOT 12x CNOT 18x CNOT Depth Circuit 72 1152 17352 Volume qubits×gates qubits×gates qubits×gates Leading- 35 p 3 (1.5E6)× p 9 (1.2E20)× p 27 order Error
Arbitrary Quantum Gates Use finite gate set from Layer 3 to approximate any arbitrary gate within precision ε Gate Sequences (no ancilla) Fowler, QIC 11 , 867-873 (2011) Gate sequence methods approximate a desired gate with fundamental gates from Layer 3 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)
Gate Sequence Methods Approximate desired gate U with some sequence of gates in fundamental set 1 0 ≈ π i 8 0 e Longer sequences produce better approximations at the expense of circuit depth and more T gates Solovay-Kitaev Fowler’s Method Phase Kickback Circuit Depth O (log c (1/ ε )) O (log(1/ ε )) RC: O ( log(1/ ε ) ) CL: O ( log(log(1/ ε )) ) 3 < c < 4 Calculation Time O (poly(log(1/ ε ))) O (poly(1/ ε )) O (1) [negligible]
Solovay-Kitaev is Expensive
Resource Analysis for Arbitrary Gates • Solovay-Kitaev appears to never produce an advantageous sequence • Fowler’s method requires exhaustive search (dashed lines extrapolated)
Separation in Time Scales Operation times increase by orders of magnitude from Physical to Logical layer
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 = 9×10 -3 Depth d = 35 Fixed size: 10 5 logical qubits • Algorithm stalls when distillation is not fast enough • Require ~90% of QC devoted to distillation
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 , 18681 (2008)] ε V = 10 -3 / ε thresh = 9×10 -3 Depth d = 31 1000 simulated time steps
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 , 735-750 (2011)] ε V = 10 -3 / ε thresh = 9×10 -3 Depth d = 31, 31, 45 (different traces) 1000 simulated time steps
Conclusions A layered architecture framework facilitates the design of fault-tolerant quantum computers The overhead costs associated with fault- tolerance 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:1010.5022] ◦ “Simulating chemistry efficiently on fault-tolerant quantum computers” [in preparation]
Auxiliary Slides
Layered Architecture
“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
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
S = exp( i π /4 σ z ) Phase Gate without Measurement 1 0 S-gate without measurement: = 0 i Still requires an ancilla state (which must be injected and distilled) However , this ancilla can be re-used
Quantum Dot Architecture Experimental Apparatus
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:
Phase Kickback in Simulation
Phase Kickback w/ Carry-Lookahead
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