Design of a Superconducting Quantum Computer: Surface Code...
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John Martinis UC Santa Barbara Design of a Superconducting Quantum Computer: Surface Code & Fidelity 1. Why surface code design? 2. Surface code review 3. Cell design methodology 4. c matrix 5. Design comments
Transcript of Design of a Superconducting Quantum Computer: Surface Code...
John Martinis
UC Santa Barbara
Design of a Superconducting
Quantum Computer:
Surface Code & Fidelity
1. Why surface code design?
2. Surface code review
3. Cell design methodology
4. c matrix
5. Design comments
Superconducting Qubits and Surface Code
3D transmons:
• T1,T2 ~ 60 ms
• tgate1 = 15 ns:
fidelity = 99.8%
• tgate2 = 200 ns:
fidelity = 99%
• Pmeas = 99.4%;
1000‟s meas./T1
• Scalable?
tgate ~ 25ns
tth ~ 2.5us
x100
quIC‟s, UCSB strategy:
• Interconnectivity
• Theory: tCZ = 25 ns,
intrinsic fidelity = 99.99%
• UCSB Xmon: T1 ~ 42 ms
Xmon
Bus
Readout
Multiplexed Measurement
Check Resonators
Broken “design rules”:• Transmon outside of
resonator• Direct mwave drive• Single ended:
galvanically connected(SQUID with high M)
(No IDC, 3um gap)(C not double-evap.)
Cbus
Creadout
Cmw
Flux bias
junctions (SQUID)
Ctransmon
UCSBXmons
200μm
J F M A M J J A S O N D J
1
10
20
50
T1 (
ms)
1.6 ms„Xmon‟
design
dispersive RO,
simple fab
MBE Al wider
CSQ
2012
CSQ
2013
3.4 ms
9.6 ms
18 ms
2012 2013
42 ms
12 msimproved
mutual
Timeline of T1 coherence
IDC transmon
phase qubit RO
Xmon yield: 26/26
Qubit freq. to 100 MHz
For next run, expect:
70 ms best, 40 ms avg.
Phase of single-shot time traces
time [us]
ph
ase
[rad
]
Quantum jumps
Resonator
Xmon qubit, |0> or |1>
input signal dispersed signalParameter Value
g/2p 30 MHz
K 1/500 ns
Δ/2p 1 GHz
<n> 160
•Measurement time: Tmeas = 800 ns•High separation gives 100% intrinsic fidelity•Errors from T1 only: Tmeas/T1 = 0.05•Surface code: k=1/50 ns possible
UCSB single shot readout with in-house paramp
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ZaZbZcZd
Measurement Symbol Physical Logic Sequence
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XaXbXcXd
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I I
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b c
d
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+-
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Zabcd =
Xabcd =
|g
repeat
+1: |yz+
-1: |yz-
+1: |yx+
-1: |yx-
|g
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Surface Code Hardware
data qubits
measure qubits
4-bit parity
4-phase parity
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Stabilized State and Identifying Qubit Errors
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All measurements X and Z commute:
Measurement outcomes unchanging
When errors:
Data qubit errors – pairs in space
Measure errors – pairs in time
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Stabilized State and Identifying Qubit Errors
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When large density -
Backing out errors not unique
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Error probability p
repeat
H H|g +-p
Model: total error p each step & qubit
Logical Error Probability
10-4
10-3
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10-5
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10-2
10-1
100
Error probability p
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Simulation:de
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ed
thL ppp / 05.0
01.0thp
measured error chain:
computed error of data qubits:
ed
ee
L pdd
ddp )8(
!)!1(
!
-
Simple statistics:
3)8(123
345pPL
E
logical error
improves with size
3p
2p
Lp
1: p/3 for X,Y, Z
2: p/15 for XX,
YY, IX, ZI …
ppppp
pp
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pppp
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p p
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The Problem: Characterizing Errors
Quantum
Process Tomography Pauli Gate Errors Randomized Benchmark
Pauli error after every gate
One & two qubit errors
Need 3 to 15 probabilities
How to measure?
Scalable
One average fidelity
Overly simple
Complex
Not scalable
Sensitive to calibration
c matrix is overly complex
Designing for the Surface Code
1/f10tgate tcycle terr-detect tlogic
qubit & gate physics measure classical logic
“clock”x8 x30
surface code design:
quantum physics prob.‟s
pX, pY, pZ
error ident.
PL
p‟s are diagonal matrix elements of
computed or measured QPT
measure(Clifford)
Theory Example: T1,T2 errors
I, X
Y, Z=
pX, pY, pZ
3 possible
probability errors:
TTT
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+
-
---
---
+
+
1
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/
11
)/(2/*
01
)/(2/
01
/
111TtTtTt
TtTtTt
eee
eee
ZZpYYpXXp
ppp
ZYX
ZYX
+++
---
)1(
+
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tp
T
tpp
step
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step
YX
is for correlated
phase errors0
tstep
Experiment Example: cerr of iswap1/2 gate
= ide
al
I, X
Y, Z
I, X
Y, Z
pXI, pYI, pZI
pIX, pIY, pIZ
pXX, pXY, pXZ
pYX, pYY, pYZ
pZX, pZY, pZZ
15 possible
probability errors:
cmeas = c+ideal cerr cideal
Recerr
Simple: Fidelity = cII,II
Modeling: Pauli errors = diagonal
Bialczak et al.
long arrows: pure dephasingshort arrows: energy relax.
S=20 MHzT1=90 ns
T2=60 ns
Correlated (k=0.5)
pure dephasing
arrows: correlation
k=0.5T=90 ns
Actual experiment(modified Pauli basis)
puredephasing
dephasingcorrelation(very small)
T extracted from QPT is closeto independently measured value
Kofman & Korotkov, arXiv:0903.0671
Local T1 & T2
energyrelaxation
Using cerr Matrix to Test for Errors
Designing for the Surface Code
1/f10tgate tcycle terr-detect tlogic
qubit & gate physics measure classical logic
“clock”x8 x30
surface code design:
quantum physics prob.‟s
pX, pY, pZ
error ident.
PL
P‟s are diagonal matrix elements of
computed or measured QPT
full model:
measure
QPT of all gates
Perr = pXXXXX…
error ident.
PL
prob.‟s
incoherent avg.
(twirling approx.)
(Clifford)
perr = pX1 + pX1Y2 + … (order 10-3)
+ a px1px2 + … (order 10-6)
need c to compute interferences, <a>=1
cerr expansion:
Surface Code Output
1/f10tgate tcycle terr-detect tlogic
measure
“clock”x8 x30
error ident. errors
length of error chain
err
or
pro
babili
ty
1 2 3 4 5 6 7 8 9 10 11 12 13 14 151510
-15
10-10
10-5
100
(1) Statistics of errors
indicates where and
type of errors
pX, pXY, …
How identify
correlated errors? PL
d/2
(2) Higher probability
from correlated errors
Example: CZ gateNatural, fast, easy to characterize
-
1
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CZ
00 01 10 11
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01
10
11
Do not care about phases
of single and 2-qubit errors:
Just measure (small) prob‟s
to put in SC errors, e.g.
pIX, pYI, pXY, pYY …
Measure accurately phases &
coherences of diagonal elements
For accuracy & get around single qubit fidelity errors,
is it possible to measure reduced process tomography?
(Still do full QPT to ensure huge error not obscured)
Architecture Design: Surface Code Cells
1) Figure of merit: size ~ tcycle (tcycle/tcoh) fast is important
Quantum Hardware: Shor Algorithm
Size ~ (p/pth) tcycle
10ns
100ns
1d10m1s
1us
10us
100d20h2m
200M20B20T
physical qubitsgate
cycle100us
1ms
30y70d3h
run time
~ constant
Limited by modular exponentiation
Need ~40N3 Toffoli (qu-AND) gates
2048 bit number, for 0.1 of threshold:
2048 factoring size N
x3600 logical overhead
x30 A states for Toffoli
220M physical qubits (1day)
Interesting example – quantum memory :
100x coherence time, 10x cycle time
Relative size = 1
Error detection: once below threshold, want to extract information quickly
serial
parallel
algor. A gener.
15M37M31B
~ tcycle / tcoh2
Architecture Design: Surface Code Cells
1) Figure of merit: size ~ tcycle (tcycle/tcoh) fast is important
2) Global optimization: data qubit with NO decoherence, pth =99% → 97%
3) Performance does not add trivially: 1 qubit exp. + 2 qubit exp. NOT scalable
Need interconnectivity
4) Quantum bus:
The quantum bus is
the smart car
5) Use ideas from the
RezQu architecture
Conclusions
1) Compute cerr
2) Need error probabilities, diagonal elements of cerr
3) Full cerr matrix not very useful; maybe for diagnostics
4) Are there alternative forms to quantify useful fidelity
based on direct measure of error probabilities?
5) Ready to test good qubits
Tsuyoshi Yamamoto
Ted White
Andrew Cleland
Aaron O‟Connell
Matthew Neeley
Peter O‟Malley
James Wenner
Jian Zhang
Michael Lenander
Erik Lucero
Martin Weides
Matteo Mariantoni
Haohua Wang
Yi Yin
Radek Bialczak
John Martinis
Julian Kelley
(Daniel Sank)
(Yu Chen) (Josh Mutus) (Shunobu Ohya) (Andrew Dunsworth)
(Rami Barends) (Evan Jeffrey) (Jimmy Chen) (Ben Chiaro)
(Charles Neill) (Anthony Megrant)
10-2
10-1
100
101
102
103
104
105
normalized physical error p/pth
num
ber
of
qubits
nq
PL = 10-20
10-10
10-5 5
10
20
40
d=
2
1
/ 05.0+
d
thL ppp
Size of Logical Qubits
p=99.9%, nq ~ 3000
p=99.99%, nq ~ 600
99.6%
physical qubits per logical qubit
