paraty - ii quantum information workshop 11/09/2009 fault-tolerant computing with biased-noise...

Paraty - II Quantum Information Workshop 11/09/2009

Fault-Tolerant Computing with Biased-Noise Superconducting Qubits

Frederico Brito

Collaborators: P. Aliferis,

D. DiVincenzo,

J. Preskill,

M. Steffen and B. Terhal.

DF- UFPE (Brazil)

Fault-Tolerant Computing With Biased-Noise Superconducting Qubits

Outline

The physical system:

– Superconducting flux qubit

Encoding scheme for a biased-noise case:

– Dephasing Vs. relaxation

Comments

2Paraty - II Quantum Information Workshop 11/09/2009



- IBM Qubit Koch et al. PRL 96, 127001 ‘06; PRB 72, 092512 ‘05.

Oscillator Stabilized Flux Qubit

- Three Josephson Junctions, Three loops

- High-Quality Superconducting Transmission Line (Q ~ 104)

- T2 = 2.7µs (memory point); T1= ~10ns (measurement point).



Energy

f c



- Qubit Potential

L R LR

L R

Burkard et al PRB 69, 064503 ‘04;

- Coupling qubit-Transmission Line Brito et al NJP 10, 033027 ‘08



- Level Dynamics

Portal

“Par

king

” R

egim

e

“S-Line”



Adiabatic Process

R,0

L,0

S,1

S,0



- Physical sources of noise

1/f noise.

Johnson noise from resistances in the circuit.

Instrumental jitter in pulse timing and amplitude.

- DC Pulse Gates

Low-bandwidth operations.

Operations are scalable.

Leakage.

Fast gates.



Phase-Gate: exp(iz)

const

t

Both qubits

Phase: 2.75 x 10-3

Relaxation: 3.5 x 10-7

Leakage : 3.77 x 10-7

Noise characterization

Noise bias



“|+>” Measurement and Preparation Gates

Non- Adiabatic Process

Qubit A = 2 3.1 GHzQubit B = 2 ¾ 3.1 GHz

Phase: 2.75 x 10-3 2.75 x 10-3

Relaxation: 3.5 x 10-7 3.5 x 10-7

Leakage : 3.77 x 10-7 1.5 x 10-5

Noise characterization

Noise bias



- Two-qubit Gate:

- Two qubit species: different transmission lines

- Qubit-qubit mutual inductance is “always on”



- CPHASE gate – Noise characterization

0 5 10 15 20 25 30 35 t(ns)

Qubit A = 2 3.1 GHzQubit B = 2 ¾ 3.1 GHz

Phase: 1.96 x 10-3 4.6 x 10-3

Relaxation: 3.5 x 10-6 3.5 x 10-6

Leakage : 3.5 x 10-6 3.5 x 10-6

Noise bias



- The IBM qubit

The noise is highly biased.

Phase errors are stronger than all other types of errors by a factor of 103.

For all other types of errors, the dominant contribution is due to relaxation to the ground state: T1 process.

Hadamard gate: error rate as low as 0.4%.

BUT, no physical implementation of a CNOT gate with error rate better than 5%.



- The IBM qubit

Simple implementations of a logical CNOT gate have error rates of the order of (a) 1.25% and (b) 2.3%.

(a) (b)

But, those implementations break the noise asymmetry, converting phase errors into errors of other types!

- For example, a z error during the implementation of a H gate will be converted to some linear combination of a z, x, and y error.



If = , and ,

- Biased-noise Qubits Can we exploit this noise asymmetry to improve the threshold

for quantum computation?

If we use an n-qubit repetition code, a first guess would lead us to the following logical errors

:

:

;

;

= phase error prob.

= error prob.

(n=7)



So, what do we need to implement that?

– A universal set of elementary operations whose implementation induces noise that is biased towards dephasing.

– All gates must commute with so that the noise bias is

maintained.

Our quantum computer will execute NJP 11, 013061 (2009)

– The preparation of the state

– The CPHASE gate;

– And measurements in the equator of the Bloch sphere:

biased noise more balanced effective noise with str. below

effective noise witharbitrarily small str.



- Logical CNOT

Logical input

data qubits

preparation

measurement

=

=

Aliferis et al quant-ph/0710.1301

Ancilla qubit



- Logical CNOT

The logical state of each block is teleported to a new block and phase errors are corrected.

The circuit prevents the propagation of leakage errors between input and output qubits (teleportation).

Measurements with ancilla qubits must be repeated several times to correct errors.

A logical teleportation preceding every logical gate prevents leakage propagation between logical gates



- The IBM qubit:

Phase errors: 4.62 x 10-3

All other erros: 3.98 x 10-3

(n,k) = (5,7)

Logical CNOT:

An improvement by a factor of about 3 over the best alternative method we have for implementing a CNOT.

Our physical-level error rates are, in principle, very close to those needed for effective fault-tolerant quantum computation!



- Comments

Can our analysis be applied for other qubits?

– We think so!

Indeed, for most qubits, dephasing is much stronger than relaxation,

Future experiments could focus on improving T1. Provided this is achieved, dephasing noise can be suppressed by using the encoding and fault-tolerant circuits we have described here.

NJP 11, 013061 (2009); NJP 10, 033027 (2008)

paraty - ii quantum information workshop 11/09/2009 fault-tolerant computing with biased-noise...

Documents

biasednoise case

johnson noise

qubit gate

qubit species

qubitqubit mutual inductance

s memory point t1

ns measurement point

pulse timing