“Putting the Bell on Schrodinger’s Cat”
PI’s: RS Rob Schoelkopf Michel Devoret Applied Physics, Yale University Luigi Frunzio Steven Girvin Leonid Glazman Liang Jiang Mazyar Mirrahimi Postdocs & students wanted! Overview
• A few remarks on quantum circuits, then and now…
• “Putting the Bell on a cat”: efficient tomography and single-shot violation for a logical qubit CHSH for encoded qubit: Petrenko, Vlastakis, et al., submitted & arXiv (2015).
• Tracking photon parity of a cat state in real time Observing quantum jumps of photon parity: Sun, Petrenko et al., Nature 511, 444 (2014). Before we met…
Some early influences:
• Koch, PhD thesis ~ 1985
• MQT: Martinis, Devoret, Clarke,…
• Control/influence of EM environment!
Devoret, Esteve, Martinis, and Urbina, 1989.
Forerunner of circuit QED!
In AMO language: strong coupling, bad cavity limit Before we met…
… decoherence…is mainly due to the… electromagnetic environment of the circuit. …we estimate that the life-time of a Q-bit …can be longer than 100 µs.
This is sufficient to perform interesting manipulations on the quantum state… When I began… First Yale/Chalmers spectroscopy of Cooper-pair box, ca. 2000 -2002, Linewidth? aka Bouchiat + RF-SET ~ 1 GHz We’ve come a long way together…
A 3D transmon qubit T1 = 103 µs
111 = +
Readout Voltage (mV) Readout Voltage T212 TTφ time (µs)
E T2 = 145 µs echo TT21~ 1.5
Readout Voltage (mV) Readout Voltage time (µs) Coherence of Photon States in Cavity
State preparation by SNAP: Heeres et al., arXiv:1503.01496, Thy: Krastanov et al., arXiv: 1502.08015
T1c = 1.2 ms T2c = 0.8 ms
Tφc = 1 ms
∆f / f = 100 Hz / 4 GHz ~ 25 ppb !! Reagor, Pfaff, et al., in preparation Dephasing and relaxation actually limited by qubit… (reverse Purcell and qubit thermal population) “Putting the Bell” on Schrodinger’s Cat
courtesy University of Illinois, Urbana-Champaign
“But who will volunteer to place it?”
Violation of a CHSH inequality “The mice in council” for a macroscopic quantum state Gustave Dore, ca. 1868 Petrenko, Vlastakis, et al., submitted (2015). Schrodinger’s Cat = an entangled state between a microscopic object (atom or qubit) and a macroscopic object (easily distinguished by “environment”)
g or e “meow?”
“ack!”
1 Ψ= ( e alive + g dead ) ?? 2 Atom: The Transmon Qubit
Superconductor ωωge≠ ef
ωef 1 nm Insulating barrier e ω
Energy ge g Superconductor (Al)
Josephson junction Non-linear ω ~ 5− 10 GHz (dissipation-free?) electromagnetic ge
oscillator ω/kB ~ 0.25 K ˆ 2 π Q 2 ˆ † †† HE= − J cos Φ =ωλge aa −aaaa +… 2C Φ0
H ≈ ωge ee Koch et al., PRA, 2007; Houck et al., PRL, 2008
Other practitioners (many!): UCSB/Google, Berkeley, Princeton, Delft, Zurich, Chicago… The Cat: A Cavity Oscillator if we can only apply classical controls (e.g. laser, force), can only make displacements
2 E = βω = nω ω E t = 0 x Glauber (coherent) state 2 2 ψ ||β xmzpf = /2 ω − ∞ β n ||β〉=en2 ∑ 〉 n=0 n! x = β aˆ β= ββ xxmax zpf What’s a Cat State of an Oscillator?
2 E = βω = nω ω E t = 0 x Cat state of an oscillator (field) 1 ψ ψ=( ββ +− ) ∆=xm /2 ω 2 Size of the superposition, d: 2 2 dn=44β = x 2 dx=2β ∆ Bigger cats die faster: rate = d κ What’s a Cat State of an Oscillator?
2 E = βω = nω ω E t = 0 x Cat state of an oscillator (field) ψ ∆=xm /2 ω 1 ψ=( ββ +− ) 2 x This one is EVEN parity! dx=2β ∆ What’s a Cat State of an Oscillator?
2 E = βω = nω ω E t = 0 x Cat state of an oscillator (field) ψ ∆=xm /2 ω 1 ψ=( ββ −− ) 2 x This one is ODD parity! dx=2β ∆ What’s a Cat State of an Oscillator?
2 E = βω = nω ω E t = 0 x
Cat state of an oscillator (field) ψ 2 ∆=xm /2 ω 1 ψ=( ββ −− ) 2 x This one is ODD parity! dx=2β ∆ What’s a Cat State of an Oscillator?
2 E = βω = nω t = π ω 2ω E
x The sign of fringe Schrödinger cat state 2 = “parity” ψ fringes ~/∆x β 1 ψ=( ββ +− ) 2 x What happens now, when packets collide? ∆=xm /2 ω Seeing the Interference: Wigner Function 2 W()α=Tr{ DD ( − αρ ) () αP} mπ mm † n Parity Pe=iπ aa =( −1)
Thy:
Negative fringes = “whiskers”
Expt’l. Wigner tomography: Leibfried et al., 1996 ion traps (NIST) Qpor Haroche/Raimond , 2008 Rydberg (ENS) Φ or x Hofheinz et al., 2009 in circuits (UCSB) Using a Cavity as a Logical Qubit? Ancilla “Hardware-efficient QEC” Readout qubits Leghtas , Mirrahimi, et al., PRL 111, 120501(2013).
High-Q (memory) Ancilla Readout qubit
Our approach : • Cavity is the memory Qubits 1..7 • One error syndrome
Register as memory: 1. 1 qubit! 1. More qubits 2. Single readout channel! 2. More decay channels earlier ideas: Gottesman, Kitaev & Preskill, PRA 64 , 012310 (2001) Encoding a Quantum Bit in a Cavity State
Continuous2-Level System Variables(e.g. transmon (e.g. cavity)) g
ge+ g+ je e Dispersive cQED Coupling: 2 Cavities + 1 Qubit + Paramp
†† H/ =−+(ωχq qsaa) e e ωs aa
FR = 98% Readout Fidelity
FP = 95% Parity Readout Fidelity cavity τµs = 55 s fs = 7.22 GHz qubit Ts1 =10µ fq = 5.94 GHz Ts2 =10µ readout τ r = 30ns f= 8.17 GHz r JBA Strong* Dispersive Regime Dispersive Hamiltonian: χ γκ 2 , χ~/g ( ωωsq− ) =ω +−ω ††χ “doubly-QND” H q ee saa a a ee interaction Allows qubit to control many photons at once (and vice-versa) n = 2 n =1 n = 0 χ n=2 n ~0 n=2 n=1 n ~ 0.5 n=1 absorption n ~1 n=0
qubit n=0
* Schuster et al., 2007; prev. attained only in Rydberg cQED (ENS-Paris) Deterministic Cat Creation: QCMAP Gate
cavity
qubit
ψ =g ⊗ 0
Theory: Leghtas et al., Phys. Rev A 87, 042315 (2013) Deterministic Cat Creation: QCMAP Gate
cavity
qubit
ψ =Ng( +⊗ e) 0
Theory: Leghtas et al., Phys. Rev A 87, 042315 (2013) Deterministic Cat Creation: QCMAP Gate
cavity
qubit
ψβ=Ng( +⊗ e)
Theory: Leghtas et al., Phys. Rev A 87, 042315 (2013) Deterministic Cat Creation: QCMAP Gate
† Hint = χaaee
cavity
qubit
ψ=Ng( ,, ββ + e e−itχ )
Theory: Leghtas et al., Phys. Rev A 87, 042315 (2013) Deterministic Cat Creation: QCMAP Gate
† Hint = χaaee after time: t = πχ
cavity
qubit ψ=Ng( ,, ββ +− e )
“Here be kittens!” Theory: Leghtas et al., Phys. Rev A 87, 042315 (2013) Encoding a Quantum Bit in a Cavity State
Continuous Variables (e.g. cavity) Efficiently Measuring , ,
Measured Wigner function of cavity 𝑐𝑐 𝑐𝑐 𝑐𝑐 Cavity state along 𝑋𝑋 𝑌𝑌 𝑍𝑍 P 2 +1 𝑋𝑋𝑐𝑐
) 𝛼𝛼 ( 0 𝐼𝐼𝐼𝐼
2 1
− − 2 0 2 ( ) − 𝑅𝑅𝑅𝑅 𝛼𝛼 Efficiently Measuring , ,
𝑐𝑐 𝑐𝑐 𝑐𝑐 Cavity state along 𝑋𝑋 𝑌𝑌 𝑍𝑍 P 2 Z X Z +1 𝑋𝑋𝑐𝑐 c c c
) 𝛼𝛼 ( 0 𝐼𝐼𝐼𝐼
2 1 Yc − − 2 0 2 ( ) − 𝑅𝑅𝑅𝑅 𝛼𝛼 Efficiently Measuring , ,
𝑐𝑐 𝑐𝑐 𝑐𝑐 Cavity state along 𝑋𝑋 𝑌𝑌 𝑍𝑍 P 2 𝑐𝑐 Zc X c Zc +1 XPc =0 = 0.76 𝑋𝑋
) 𝛼𝛼 ( 0 YP=jπ = 0.14 c 𝐼𝐼𝐼𝐼 8β
Z=−= PP0.07 2 1 c ββ− Yc − − 2 0 2 ( ) − 𝑅𝑅𝑅𝑅 𝛼𝛼 Efficiently Measuring , ,
𝑐𝑐 𝑐𝑐 𝑐𝑐 Cavity state along 𝑋𝑋 𝑌𝑌 𝑍𝑍 P 2 Z X Z +1 𝑌𝑌𝑐𝑐 c c c
) 𝛼𝛼 ( 0 𝐼𝐼𝐼𝐼
2 1 Yc − − 2 0 2 ( ) − 𝑅𝑅𝑅𝑅 𝛼𝛼 A Bell State 1 ψ B =( ge + eg) 2
2-Level System 2-Level System (e.g. transmon) (e.g. transmon) g g
ge+ g+ je ge+ g+ je The Schrodinger Cat or “Bell-Cat” 1 ψB =( ge ββ +−) 2
2-Level System Continuous (e.g. transmon) Variables (e.g. cavity) g
ge+ g+ je Measuring Bell-Cat Correlations
X P0 X C
ψ Pjπ B Y 8β YC
Z PPββ− − ZC
Vlastakis et.al. Science 2013 Performing a CHSH Measurement
OABA'' B =+− ABc ABc ' A '' Bc + A ' Bc
X P0 X C
ψ Pjπ B Y 8β YC y Rθ P X Z PP0ββ− − ZCC y Rθπ+ /2 PPββ− − ZC
Vlastakis et al. Science 2013 Violating the CHSH Inequality
OXZZX=+−+ XZ c XXc ZXc ZZc = 1 g 𝛽𝛽
45° 1 ° ( ge+ ) 135 2 Bell Signal 𝑂𝑂
e
MAX VIOLATION MAX VIOLATION Violating the CHSH Inequality
WE TAKE ALL MEASUREMENTS: No correction for detector inefficiency = 1
2.30 ± 0.04 𝛽𝛽
OXZZX
OZXXZ Bell Signal 𝑂𝑂
2.28 ± 0.04
− Continuously Varying Bell-Cat Size
e dead + g alive
= 2
𝑑𝑑 𝛽𝛽
2.14 ± 0.04 Still violating at Increasing : = 16 photons
Bell Signal 𝑂𝑂 Finite encoding and Realizing a Schrodinger’s Cat measurement fidelity 2 Experiment𝛽𝛽 𝑑𝑑
see also Brune et al., 1996 What Comes Next? Cavity as a Correctable Memory? Leghtas, Mirrahimi, et al., PRL 111, 120501(2013).
+ 0L =CNα = ()ββ +−
+ 1Li=CNα = ()iiββ +−
ccgL↓ + ccee↑ ⇒ g 0 + 1L
Store a qubit as a superposition of two cats of same parity Z. Leghtas M. Mirrahimi Cat States for Hardware-Efficient QEC? “Cat codes”: much less hardware Leghtas, Mirrahimi, et al., PRL 111, 120501(2013). required High-Q (memory) Ancilla Readout qubit
1st tracking of a parity or error syndrome in real-time: odd even odd even odd even
parity msmt. ala’ Bertet et al., Sun, Petrenko et al., Nature 511, 444 (2014). Bon anniversaire Quantronics!
Merci! … and my apologies Summary • Coherence in circuit QED passing the QEC threshold. Qubits: T2 ~ 2*T1 ~ 0.0001 sec Cavities: T2 ~ 2*T1 ~ 0.001 sec
• Cat-codes: a new shortcut for QEC?
• Tracking the jumps of an error syndrome: photon parity
Next challenge: “breakeven” for error correction!
• Bell violation between qubit and continuous variable system: benchmarking a module Leghtas, Mirrahimi et al., PRL 2013. Sun, Petrenko et al., Nature 511, 444 (2014). Petrenko, Vlastakis et al., submitted & arXiv (2015)