Performance Requirements of a Quantum Computer Using Surface Code - - PowerPoint PPT Presentation

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 2nd 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)

Physical Gate

• Stimulated Raman transition with ultrafast

broadband pulse

• Press, et al. Nature 456 218-221 (2008)

Measurement

• Dispersive optical spin measurement
• Atatüre, et al. Nature Physics 3, 101 (2007)
• Coherence time (T2 ~ 3 μs) [Press, et al. Nature Photonics 4, 367-370 (2010)]
• Gate execution times ( 20 – 40 ps)
• Errors – systematic or random?

Laser pulses

= B g h

B

µ

Virtual Layer

• Cause destructive interference of systematic errors

Dynamical decoupling sequence to correct dephasing errors

• T2

* 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)

Virtual Layer

Fowler, et al. Phys. Rev. A 80, 052312 (2009) Fowler, et al. arXiv/1110.5133 (2011)

Quantum Error Correction Layer

• Surface code: estimate distance needed

Extract Syndrome Syndrome Matching εthresh 9×10-3 εV 10-3 C 0.03 εL 10-15 d 29 Estimating Surface Code Distance

Quantum Error Correction Layer

Logical Layer

• Use fault-tolerant QEC to deliver any arbitrary gate

to the Application Layer

15 faulty ancillas 1 purified ancilla State Distillation

• Bravyi and Kitaev, Phys. Rev. A 71, 022316 (2005)

Approximating Arbitrary Quantum Gates Methods with and without 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) ~35p3

• e.g., 2 levels distillation:

[p0 = 10-3]  [p2 = 1.5 ×10-21]

Distillation Levels 1 Level 2 Levels 3 Levels

• Min. Circuit

Depth 6x CNOT 12x CNOT 18x CNOT Circuit Volume 72 qubits×gates 1152 qubits×gates 17352 qubits×gates Leading-

• rder Error

35p3 (1.5E6)×p9 (1.2E20)×p27

Arbitrary Quantum Gates

• Use finite gate set from Layer 3 to approximate any

arbitrary gate within precision ε

Gate sequence methods approximate a desired gate with fundamental gates from Layer 3 Phase kickback uses a special ancilla state to perform phase gates Although requires more qubits, can have lower circuit depth Gate Sequences (no ancilla) Phase Kickback (multi-qubit ancilla)

Fowler, QIC 11, 867-873 (2011) 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

Solovay-Kitaev Fowler’s Method Phase Kickback Circuit Depth O(logc(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] Longer sequences produce better approximations at the expense of circuit depth and more T gates

        ≈

8

1

π i

e

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)] εV = 10-3 / εthresh = 9×10-3 Depth d = 35 Fixed size: 105 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)] εV = 10-3 / εthresh = 9×10-3 Depth d = 31 1000 simulated time steps

Quantum Simulation (Second-Quantized)

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)] εV = 10-3 / εthresh = 9×10-3 Depth d = 31, 31, 45 (different traces) 1000 simulated time steps

• LiH energy eigenvalue using STO-3G basis

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(iπ/4 σz) Phase Gate without Measurement

 S-gate without measurement:  Still requires an ancilla state (which must

be injected and distilled)

 However, this ancilla can be re-used

      = i 1

Quantum Dot Architecture Experimental Apparatus

Phase Kickback (Kitaev-Shen-Vyalyi)

• Use multi-qubit ancilla for phase gate rotations

When controlled-addition is performed on the ancilla, a phase is “kicked back” to the control qubit: This ancilla is an eigenstate of addition; the eigenvalue is a phase rotation:

Phase Kickback in Simulation

Phase Kickback w/ Carry-Lookahead