Download Introduction to quantum computing (with - DiCarlo Lab

Introduction to quantum computing
(with superconducting circuits)
Lecture 2
Version: 14/08/29
Frontiers of Condensed Matter
San Sebastian, Aug. 28-30, 2014
Dr. Leo DiCarlo
[email protected]
dicarlolab.tudelft.nl
Summary of Lecture #1
θ
Quantum states
ϕ
X
Quantum gates
QFT
H
Uf
Rnˆ (θ )
Quantum msm’t
M̂
Anatomy of a simple quantum algorithm
Qubit
register
M
Ancilla
qubits
initialize
create
maximal
superposition
encode function
in a unitary
clever process
measure
will involve entanglement and
disentanglement between qubits
Maintain quantum coherence
(DiVincenzo criteria)
1) Start in superposition: all values at once!
2) Build complex transformation out of one-qubit and two-qubit gates
3) Somehow* make the answer result in a computational-basis state at end!
*use quantum interference: the magic of properly designed algorithm
Outline of Lecture #2
● Quantum computing hardware based on
superconducting circuits
One- & two- qubit gates
Quantum measurement
Simple quantum algorithms (games) achieving
speedup.
Several architectures have ran quantum algorithms
Photons
Nuclear Spins (NMR)
NV centers
1998
Trapped Ions
2003
2000
2012
Superconducting
circuits
2009
The appeal of integrated electrical circuits
Modular architecture: few building blocks
Intel
Parallel fabrication
LC circuit as a quantum harmonic oscillator
E
φ
7
6
+Q
L
5
C
4
3
-Q
2
ωr
(
Hˆ ωr aˆ † aˆ + 1
=
2
)
φˆ
Qˆ
φˆ
Qˆ
†
aˆ =
+i
−i
; aˆ =
φZPF QZPF
φZPF QZPF
φZPF
=
2ωr L ; QZPF
2ωr C
1
0
φ
annihilation and creation operators
 aˆ , aˆ †  = 1
trapped photons!
M. Devoret, Les Houches Session LXIII (1995)
Why microwave frequencies + cryogenic temp?
CAN PLACE CIRCUIT IN ITS QUANTUM GROUND STATE
E
7
6
5
4
3
2
ω r
1
0
φ
 ω r  k BT
P(n = 0) = 1 − e
5 GHz
20 GHz ~ 1 K
− ω r / k B T
Why superconducting?
φ
E
7
6
5
4
3
2
ωr
important: as little
dissipation as possible
1
0
φ
dissipation broadens energy levels,
enhancing energy relaxation
Why Josephson?
E
7
6
generator = µwave LASER
5
4
3
ωr
2
1
0
φ
IN LINEAR CIRCUITS, ALL TRANSITIONS BETWEEN QUANTUM
LEVELS ARE DEGENERATE!
Cannot confine system to a two-level subspace = LEAKAGE
Why Josephson?
NEED NONLINEARITY TO FULLY REVEAL QM
E
Position coordinate
Emission
spectrum
ω34 ω23 ω12 ω01
frequency
Josephson tunnel junction
A NON-LINEAR INDUCTOR WITH NO DISSIPATION
Ι
~1nm
S
I
S
superconductorinsulatorsuperconductor
tunnel junction
φ = ∫−∞ V ( t ')dt '
t

φ0
φ 
I=
sin  2π 
2π LJ
 φ0 

φ 
U = − EJ cos  2π 
 φ0 
φ0 =
2 EJ
bare Josephson potential
φ
h
2e
Josephson junctions in real life: imperfectly beautiful
Al/AlOx/Al
credit L. Frunzio and D. I. Schuster
credit I. Siddiqi and F.Pierre
EJ ~ 50 K
LJ ~ 150 pH
100nm
EJ ~ 0.5 K
LJ ~ 15 nH
Superconducting circuits: artificial atoms
Josephson
junctions
C
E
3
2
1
0
φ
Transmon qubit: Koch et al., PRA (2008), Schreier et al., PRB (2008)
Flux control of qubit frequencies
I1
Φ1
Fast
& Local
LJ (Φ )
C
I2
Φ2
ω01 / 2π ≈
1
~ 4 − 8 GHz
LJ (Φ )C
(ω01 − ω12 ) / 2π ≈ 300 MHz
Cavity QED with wires
“Circuit QED”
Wallraff et al., Nature (2004)
Blais et al., Phys. Rev. A (2004)
out
Josephson-junction
qubits
in
Transmission-line resonator
• mediates interaction between qubits
• protects qubits from continuum
• allows qubit readout
Expts: Sillanpää et al., Nature (2007)
Majer et al., Nature (2007)
(Phase qubits / NIST)
(Transmon qubits / Yale)
First quantum processors
2009 model
DiCarlo et al., Nature (2009)
1 mm
2010 model
DiCarlo et al., Nature (2010)
Reed et al., Nature (2012)
2012 model
J. P. Groen et al., Phys. Rev. Lett. (2013)
2013 model
O.P. Saira et al., Phys. Rev. Lett. (2014)
Latest Delft processor (2014 Model)
2014 Model: 2D+ connectivity
flux controls
feedline
Control and readout by
frequency multiplexing
CPW cross-overs
Air bridges
Similar developments at Chalmers, UCSB, ETH
3D circuit QED
200 nm
250 µm
A Schoelkopf group breakthrough: H. Paik et al., PRL (2011)
A quantum computing roadmap
Review: Devoret & Schoelkopf, Science (2013)
Single-qubit control and measurement
0
X =
1
1
0

=
X
0
Y =
i
−i 
0

=
Y
Rotations
1
Z =
0
0 
−1

=
Z
 
Rˆ nˆ (θ ) = cos(θ / 2) Iˆ − i sin(θ / 2)n ⋅ σ

σ = Xˆ , Yˆ , Zˆ
H =
1
2
1
1

1 
−1

=
H
Rn (θ )
{
}
One-qubit gates: X and Y rotations
fL
z
Preparation
1-qubit rotations
Measurement
y
x
cavity
I
cos(2π f Lt )
V
R transmon (a.u.)
Flux bias on right
One-qubit gates: X and Y rotations
fR
z
Preparation
1-qubit rotations
Measurement
y
x
cavity
I
cos(2π f R t )
V
R transmon (a.u.)
Flux bias on right
One-qubit gates: X and Y rotations
fR
z
Preparation
1-qubit rotations
Measurement
y
x
cavity
Q
sin(2π f R t )
V
R transmon (a.u.)
Flux bias on right
see Fidelity > 99%
J. Chow et al., PRL (2009)
Individual qubit readouts
NbTiN
Coupling resonator - QUANTUM BUS
Readout Q1
Readout Q2
Qubit 1
Qubit 2
2 mm
Feedline
Readout 1
Readout 2
01
00
10
00
11
10
11
01
Characterizing individual readout fidelity
NbTiN
Readout 1
Readout Q2
10
00
11
01
Qubit 1
averaged transients
1i
0i
Qubit 2
Feedline
2 mm
single-shot histograms
Characterizing individual readout fidelity
NbTiN
Readout 1
Readout Q2
10
00
11
01
Qubit 1
averaged transients
1i
0i
Qubit 2
Feedline
2 mm
single-shot histograms
Characterizing individual readout fidelity
NbTiN
Readout 1
Readout Q2
10
00
11
01
Qubit 1
Qubit 2
2 mm
Feedline
error budget
0i
96%
+1
Fidelity = 84%
1i
88%
-1
A quantum computing roadmap
C-NOT/C-Phase, Deutsch-Jozsa, Grover’s,
Measurement-free error correction (repetition code)
Review: Devoret & Schoelkopf, Science (2013)
A universal set of gates
0
X =
1
1
0

=
X
0
Y =
i
−i 
0

=
Y
Rotations
1
Z =
0
0 
−1

=
Z
 
Rˆ nˆ (θ ) = cos(θ / 2) Iˆ − i sin(θ / 2)n ⋅ σ

σ = Xˆ , Yˆ , Zˆ
1
0
U =
0

0
0
1
0
0
0 0
0 0

1 0

0 −1
=
Conditional phase gate
1
H =
2
1
1

1 
−1

=
H
Rn (θ )
{
1
0
U =
0

0
0
1
0
0
0
0
0
1
0
0

1

0
Controlled-NOT
}
Spectroscopy of two qubits + cavity
VR
right qubit
Dispersive
qubit-qubit
swap interaction
left qubit
Cavity-qubit interaction
Vacuum Rabi splitting
cavity
V
R transmon (a.u.)
Flux bias on right
Background:
Majer et al., Nature (2007)
Wallraff et al., Nature (2004)
Resonant qubit-bus interaction
right qubit
left qubit
Cavity-qubit resonant interaction
Vacuum Rabi splitting
cavity
2g
V
R transmon (a.u.)
Flux bias on right
2g
Background:
Majer et al., Nature (2007)
Wallraff et al., Nature (2004)
Dispersive qubit-qubit interactions
2g × ( g / ∆)
right qubit
natural
speed
Dispersive
qubit-qubit
swap interaction
slow-down
factor ~1/10
2g / ∆
2
left qubit
∆
cavity
V
R transmon (a.u.)
Flux bias on right
Background:
Majer et al., Nature (2007)
Wallraff et al., Nature (2004)
Two-qubit gate: turn on interactions
VR
Conditional
phase gate
Use control lines to push
qubits near a resonance
cavity
V
R transmon (a.u.)
Flux bias on right
Two-excitation manifold of system
20
• Transmon “qubits” have
multiple levels…
11
Two-excitation
manifold
• Avoided crossing (160 MHz)
11 ↔ 20
Flux bias on right transmon (a.u.)
Strauch et al. PRL (2003):
proposed using interactions with higher levels for
computation in phase qubits
Adiabatic conditional-phase gate
20
11
tf
f 01 + f10
ϕa = −2π ∫ δ f a (t )dt
t0
2-excitation
manifold
ζ
11 → eiϕ11 11
tf
ϕ11 = ϕ10 + ϕ01 − 2π ∫ ζ (t )dt
t0
10
1-excitation
manifold
10 → e
01
01 → e
Flux bias on right transmon (a.u.)
iϕ01
iϕ10
10
01
Implementing C-Phase with 1 fancy pulse
00
01
10
11
1 0
 0 eiϕ01
Uˆ  
0 0

0 0
0
0  00
0  01

0  10
iϕ11  11
e 
0
eiϕ10
0
Adjust timing of flux pulse so that
only quantum amplitude of 11
acquires a minus sign:
1
0
Û  
0

0
0
1
0
0
0 0
0 0

1 0

0 −1
π
Gates at the raw speed of circuit QED
Q
B
g D 21B ↔ eD 2 0 B
swap in
in 10 ns
02
11
11
01
10
10
20
20
eD 21B ↔ g D 2 2 B
eD 21B ↔ f D 2 0 B
c-phase in
in <20 ns
00
Saira et al., PRL 112, 070502 (2014)
Proposed by: G. Haack et al., PRB (2010)
Gates at the raw speed of circuit QED
Q
B
g D 21B ↔ eD 2 0 B
swap in
in 10 ns
eD 21B ↔ g D 2 2 B
02
11
01
10
20
eD 21B ↔ f D 2 0 B
c-phase in
in <20 ns
00
Saira et al., PRL 112, 070502 (2014)
Proposed by: G. Haack et al., PRB (2010)
Gates at the raw speed of circuit QED
Q
B
g D 21B ↔ eD 2 0 B
swap in
in 10 ns
−1
eD 21B ↔ g D 2 2 B
02
01
11
20
eD 21B ↔ f D 2 0 B
c-phase in
in <20 ns
10
00
Saira et al., PRL 112, 070502 (2014)
Proposed by: G. Haack et al., PRB (2010)
Generating and detecting 2-qubit entanglement
0
0
Rπy /2
Rπy /2
±1
±1
π /2
Ry
ρ=
Z
∑
σ kσ j
j ,k∈{ I , X ,Y , Z }
4
σ kσ j
Z
Pauli set
00
01
10
11
11 00 01 10
Tomography with joint readout: Filipp et al., PRL (2009)
Bell inequalities
z
CHSH = X ′X + X ′Z − Z ′X + Z ′Z
x’
z’
CHSH = X ′X − X ′Z + Z ′X + Z ′Z
θ
x
no readout correction
Clauser, Horne,
Shimony & Holt (1969)
LHV bound:
CHSH ≤ 2
1.80 ± .01
UCSB group has closed detection loophole (w/ 2.07): Ansmann et al., Nature (2009)
Bell inequalities
z
CHSH = X ′X + X ′Z − Z ′X + Z ′Z
x’
z’
CHSH = X ′X − X ′Z + Z ′X + Z ′Z
θ
x
Clauser, Horne,
Shimony & Holt (1969)
LHV bound:
CHSH ≤ 2
not a foolproof
test of
hidden variables…
(locality & detection
loopholes)
With joint readout: Chow et al., PRA (2010)
with readout correction
2.57 ± .01
Deutsch’s problem: is your coin fair?
f2
f1
f3
f4
0
0
0
0
0
0 0
0
1
1
1
1
1
1 1
1
unbalanced
balanced
Classical Problem:
You are handed a “black box” with one of the functions f i
programmed in, but you’re not told which one.
Determine if the function is balanced or unbalanced.
Deutsch’s quantum algorithm
● Execute this sequence calling the quantum black box once.
0
1
H
x
H
y
Uf
x
H
m
Ẑ
y ⊕ f ( x)
m = +1
function is unbalanced
m = −1
function is balanced
Also implemented in
NMR: Chuang et al., Nature (1998)
Ion traps: Guide et al., Nature (2003)
NV centers: Van der Sar et al., Nature (2012)
Encoding of the functions
f1
f4
f3
f2
0
0
0
0 0
0 0
0
1
1
1
1 1
1 1
1
U f1
1
0

0

0
0
1
0
0
0
0
1
0
0
0

0

1
0
1

0

0
1
0
0
0
0
0
0
1
x
y
U f3
U f2
X
0
0

1

0
1
0

0

0
0
1
0
0
0
0
0
1
U f4
0
0

1

0
0
1

0

0
1
0
0
0
0
0
1
0
0
0

0

1
85%
85%
86%
86%
Single-shot success
Quantum speedup in Deutsch-Jozsa algorithm
speedup
no speedup
-1 1
-1 1
-1 1
-1 1
Quantum
speedup
This work: J. Cramer Master’s thesis, TU Delft (2012)
First demonstration of quantum speedup in sc. circuits: Yamamoto et al., PRB (2010)
Grover’s search algorithm
● Consider the n-to-1 bit function:
∗
y f=
( x)
=
● Problem: find
x
1 for x = x
∗
0 for x ≠ x
∗
Case n=2:
Classically, takes on average 2.25 uses of the black box
to succeed…
Quantum mechanically,
1 use of the quantum black box gives right answer!
Grover’s search algorithm
● Execute this sequence, which calls the quantum black box once.
0
H
x
0
1
x
H
H
● Answer:
Uf
y
H
π
H
m1
Ẑ
m0
H
H
Ẑ
Grover’s analysis
“inversion about mean”
y ⊕ f ( x)
(+1, +1) x = 00
(+1, −1)
∗
= 01
x

=
⇒ ∗
(m1 , m0 ) 
x = 10
(−1, +1)
(−1, −1)
x∗ = 11
∗
Grover’s search algorithm
0
H
H
x
0
H
1
H
x
H
Uf
y
π
H
m1
Ẑ
m0
H
Ẑ
y ⊕ f ( x)
Quantum phase kick-back
Grover’s search algorithm
x∗
0
0
H
H
1
0

0
0

0 0
1 0
0 −1
0 0
00 + 01 + 10 + 11
0
0

0
1 
H
H
π
H
m1
Ẑ
m0
H
Ẑ
00 + 01 − 10 + 11
− 00 − 01 + 10 + 11
−( 0 − 1 ) ⊗ ( 0 + 1 )
− 10
0
0
Grover, PRL (1997)
Rπy /2
Rπy /2
Qoogle
Grover as a quantum game between our ‘twins’
Rπy /2
Rπy /2
π
Rπy /2
10
Rπy /2
Quantum searching algorithm step-by-step
ψ ideal = 00
Rπy /2
0
b
0
Rπy /2
c
Qoogle
Begin in ground state:
10
Rπy /2
d
e
Rπy /2
π f
Rπy /2
Rπy /2
g
DiCarlo et al., Nature (2009)
Quantum searching algorithm step-by-step
1
ψ ideal= ( 00 + 01 + 10 + 11 )
2
Rπy /2
0
b
0
Rπy /2
c
Qoogle
Create a maximal
superposition:
look everywhere
at once!
10
Rπy /2
d
e
Rπy /2
π f
Rπy /2
Rπy /2
g
Quantum searching algorithm step-by-step
1
ψ ideal= ( 00 + 01 − 10 + 11 )
2
Rπy /2
0
b
0
Rπy /2
c
Qoogle
Apply Qoogle to
mark the solution
10
Rπy /2
d
e
Rπy /2
π f
Rπy /2
Rπy /2
g
Quantum searching algorithm step-by-step
1
ψ ideal
00 + 11 )
(
2
Some more 1-qubit
rotations…
Rπy /2
0
b
0
Rπy /2
c
Qoogle
Now we arrive in
one of the four
Bell states
10
Rπy /2
d
e
Rπy /2
π f
Rπy /2
Rπy /2
g
Quantum searching algorithm step-by-step
1
ψ ideal= ( 00 − 01 + 10 − 11 )
2
Rπy /2
0
b
0
Rπy /2
c
Qoogle
Another (but known)
2-qubit operation now
undoes the entanglement
and makes an interference
pattern that holds
the answer!
10
Rπy /2
d
e
Rπy /2
π f
Rπy /2
Rπy /2
g
Quantum searching algorithm step-by-step
ψ ideal = 10
Final 1-qubit
rotations reveal the
answer:
Focus quantum
amplitude on the
answer: “10”!
Rπy /2
0
b
0
Rπy /2
c
Qoogle
Correct answer would
found >80% of the time!
10
Rπy /2
d
e
Rπy /2
π f
Rπy /2
Rπy /2
g
0
0
Rπy /2
Rπy /2
oracle
Quantum speedup in 2-qubit Grover algorithm
ij
π
Rπy /2
m1
Rπy /2
m2
Rπy /2
0
Rπy /2
0
oracle
Quantum speedup in 2-qubit Grover algorithm
π
ij
1
0
Rπy /2
m1
Rπy /2
m2
84% = Psuccess, perfect readout 84%
87%
87%
-1
00
01
10
11 00 01 10 11
Essentially the same algorithmic fidelity as we did in 2009:
DiCarlo et al. Nature (2009) (Yale)
Rπy /2
0
π /2
0
Ry
oracle
Quantum speedup in 2-qubit Grover algorithm
π
ij
m1
π /2
Quantum
speedup
87%
m2
Ry
1
84% = Psuccess, perfect readout 84%
87%
0
-1
Rπy /2
This work: J. Cramer Master’s thesis, TU Delft (2012)
First demonstration of quantum speedup in sc. circuits: Dewes et al., PRL & PRB (2012)
(52-67% success)
speedup
71%
71%
74%
72%
Single-shot success
00
01
10
11 00 01 10 11
no speedup
00 01 10 11
00 01 10 11
00 01 10 11
00 01 10 11
Grover search beyond 2 qubits
Grover’s iteration
0
H
H
x
0
1
H
H
x
Uf
y
y ⊕ f ( x)
Case n>2: by computer demo.
H
π
H
m1
Ẑ
m0
H
Ẑ
Grover’s analysis
“inversion about mean”
Demo on computer
Performance of Grover’s quantum algorithm
O ( 2n−1 )
O
( N)
O ( 2n/2 )
● Grover’s algorithm does not scale polynomially with the number of bits n,
and hence it is not efficient!
● The best classical approach requires, on average, N/2=2n-1 calls of the classical
black box.
● The quadratic speedup offered by Grover’s is still useful.
Summary of Lecture #2
● Superconducting quantum processors based on circuit quantum electrodynamics:
Nonlinear LC oscillators as qubits;
Interconnected, readout and protected using resonators.
● Universal gate set based on single-qubit rotations and C-Phase gates.
● Can controllably create and undo entanglement.
● Simple quantum games (Deutsch-Josza, Grover’s) with quantum speedup achieved.
● The power of a quantum algorithm lies in using quantum superposition to create all
possible inputs at once, and then evaluating a function for all inputs in one call!
● The challenge in designing a quantum algorithm lies in finding an analysis step that
uses quantum interference to focus quantum probability amplitudes toward the
solution of the problem.
● Some computational problems can be solved more efficiently using a quantum
computer.
Specific example: Classical & Quantum seearch
scaling
N vs
N
Tomorrow: basic quantum error correction (thy and expt)