Guest lecture by Dr. Arne Grimsmo

Quantum Mechanics

Lecture 19 (non-examinable)

Cat states;Encoding quantum information in harmonic oscillators.

Quantum harmonic oscillators in real life

Mechanical

500

Electrical

Electromagnetic fields in cavities

4.5. CYLINDRICAL CAVITY RESONATORS 100

SMA teeRF in RF out

coupler

E H

(a) (b)

field10

Figure 4.9: Cylindrical TE011 resonator. (a) Field distribution of the TE011 mode. Theelectric field is concentrated in a torus about the cylindrical zaxis, and the electric fieldvectors point along �. It is visible on the plot that the electric field goes to zero on all thecavity wall. The magnetic field is zero at the the corners of the cavity and a maximum at thecenter. (b) Physical realization of the cylindrical cavity. Two lids close the cylindrical cavityalong the region of minimum magnetic field. Additionally, the small ring perturbation on thelids, which detunes the degenerate TM111 modes, is visible. An SMA tee allows this cavityto be measured with the shunt technique. The coupling from SMA to cavity is accomplishedvia a loop coupler as described in Section 4.5.2. (Figure used with permission from [100].See Copyright Permissions.)

degeneracies at this small of a limit. The above estimates serve as a lower bound then forthe splitting between modes. Experimentally, we observe a detuning between TE011 andTM111 modes of �!/2⇡ ⇡ 30MHz.

The other modes of the cylinder are useful to study many dissipation mechanisms atonce, since they will all have various sensitivities to materials and assembly. For instance,if the TE011 mode is significantly higher Q than any of these other modes, we require lossmechanisms beyond conductor loss to explain the other modes (see Table 4.2) [100].

4.5.2 Input-output coupling

As shown in Figure 4.10, coupling to the cylindrical cavity is achieved in a similar manneras before (see Sec. 4.4.2). However, due to the special configuration of the TE011 mode,magnetic dipole coupling is required here. Additionally, several other considerations are

LC

Quantum phase space and Wigner functionsOne way to make sense of quantum phase space is with the Wigner function. Starting from a wave function ψ, we transform it as follows:

W(x, p) =1

πℏ ∫∞

−∞⟨x − y |ψ⟩⟨ψ |x + y⟩ e2iyp/ℏ dy

W(x,p) can be used to visualize quantum states

�2 0 2x

�2

0

2

p

�0.4000�0.3192�0.2384�0.1576�0.07680.00400.08480.16570.24650.3273

�2 0 2x

�2

0

2

p

�0.4000�0.3192�0.2384�0.1576�0.07680.00400.08480.16570.24650.3273

|α = 2⟩ |α⟩ + | − α⟩

�2 0 2x

�2

0

2p

�0.4000�0.3192�0.2384�0.1576�0.07680.00400.08480.16570.24650.3273|n = 5⟩

“cat state”number statecoherent state

Bits and qubits: From classical to quantum information

A single bit takes a binary value {0, 1} and is the “unit” of classical information

Is there a quantum analog, a “unit” of quantum information?

{ |0⟩, |1⟩} c0 |0⟩ + c1 |1⟩

In principle any pair of quantum states will do

For example, the two lowest states of a quantum harmonic oscillator

|n = 0⟩, |n = 1⟩0

|0

|1

|2

V(x

)[arb.units]

Lorem ipsum

ω

The “qubit”:

Manipulating classical information

Any classical computation can be generated using only a very small set of “logic gates” acting on 1 or 2 bits at a time, e.g.

NOT

AND

Manipulating quantum information

Quantum “logic gates” are unitary operators acting on a state

U |ψ⟩ = U (c0 |0⟩ + c1 |1⟩)

U |ψ⟩ = U (c00 |0⟩ |0⟩ + c01 |0⟩ |1⟩ + c10 |1⟩ |0⟩ + c11 |1⟩ |1⟩)

One-qubit gate:

Two-qubit gate:

Example:

(2x2 matrix)

(4x4 matrix)

What are quantum computers good for?

Quantum computers can (in theory) solve in a matter of days/hours/weeks some computational problems that would take a conventional computer longer than the lifetime of the universe. The potential is vast…

Chemistry and material simulations (drug discovery, new materials etc.)

Optimization problems

Machine learning

Save the environment, cure cancer, end poverty, …

Why don’t we have (useful) quantum computers yet?

The fundamental problem: qubits are extremely fragile

0

|0

|1

|2

V(x

)[arb.units]

Lorem ipsum

Energy loss a(c0 |0⟩ + c1 |1⟩) = c1 |0⟩

Encoded information is lost!

aIn practice, many other sources of errors, but energy loss is often the dominant cause of faults.

Encoding quantum information robustlyIdea: Use coherent states as logical states

|0⟩ → |α⟩, |1⟩ → | − α⟩

a | ± α⟩ ∝ ± |α⟩States not destroyed: Good!

But superpositions not preserved:

a(c0 |α⟩ + c1 | − α⟩) ∝ c0 |α⟩ − c1 | − α⟩c1 → − c1

this is called a “phase error” Bad!

So this did not quite work… can we fix it?

�2 0 2x

�2

0

2

p

�2 0 2x

�2

0

2

p

|α⟩ | − α⟩

Cat-state qubitsNew idea: Use superpositions of coherent states as logical states

|0⟩ → |0L⟩ = |α⟩ + | − α⟩

States not destroyed, but they do change:

|1⟩ → |1L⟩ = | iα⟩ + | − iα⟩

a |0L⟩ ∝ |α⟩ − | − α⟩ =: |0′�L⟩

a |1L⟩ ∝ | iα⟩ − | − iα⟩ =: |1′�L⟩

a(c0 |0L⟩ + c1 |1L⟩) ∝ (c0 |0′�L⟩ + c1 |1′�L⟩)

Quantum information in principle preserved, even though the states changed!

�2 0 2x

�2

0

2

p

�2 0 2x

�2

0

2

p

|α⟩ + | − α⟩ | iα⟩ + | − iα⟩

|0⟩ → |0L⟩ = |α⟩ + | − α⟩ |1⟩ → |1L⟩ = | iα⟩ + | − iα⟩

a(c0 |0L⟩ + c1 |1L⟩) ∝ (c0 |0′�L⟩ + c1 |1′�L⟩)

If we can find a way to detect that the error has happened, we can simply update the “encoding” |0L⟩, |1L⟩ → |0′�L⟩, |1′�L⟩

But how do we do this without destroying the encoded information?

What does the measurement need to distinguish, and what must it not distinguish?

Cat-state qubits

Detecting errors

Let’s have a look at what the states look like in the number basis

|0L⟩ = |α⟩ + | − α⟩ = Cα

∞

∑n=0

αn + (−α)n

n!|n⟩ = 2Cα

∞

∑n=0

α2n

(2n)!|2n⟩

|0′�L⟩ = |α⟩ − | − α⟩ = Cα

∞

∑n=0

αn − (−α)n

n!|n⟩ = 2Cα

∞

∑n=0

α2n+1

(2n + 1)!|2n + 1⟩

|α⟩ = Cα

∞

∑n=0

αn

n!|n⟩Recall:

Similarly

|0L⟩ = 2Cα

∞

∑n=0

α2n

(2n)!|2n⟩

|1L⟩ = 2Cα

∞

∑n=0

(iα)2n

(2n)!|2n⟩

|0′�L⟩ = 2Cα

∞

∑n=0

α2n+1

(2n + 1)!|2n + 1⟩

|1′�L⟩ = 2Cα

∞

∑n=0

(iα)2n+1

(2n + 1)!|2n + 1⟩

Odd number parity

Measure number parity to detect error! Is there an observable for number parity? Π = (−1) n = (−1) a† a

Detecting errors

Even number parity

1. Encode

2. Check for errors by measuring number parity

3. Re-define encoding as needed if parity has changed

4. Repeat 2. & 3. for as long as we need to store the information

A protocol for storing quantum information robustly

|0⟩ → |0L⟩ = |α⟩ + | − α⟩ |1⟩ → |1L⟩ = | iα⟩ + | − iα⟩

|0L⟩ → |0′�L⟩ |1L⟩ → |1′�L⟩

Π = (−1) n = (−1) a† a

0 0 M O N T H 2 0 1 6 | V O L 0 0 0 | N A T U R E | 3

LETTER RESEARCH

the average photon number in the resonator, starting at tw ≈ 15 µ s and increasing after each subsequent step up to tw ≈ 25 µ s to account for the decay of the average photon number. Without post-selection, the cat code outperforms the uncorrected transmon with a time constant of exponential decay that is a factor of about 20 higher, indicating that although the coupling between the resonator and transmon is always

on, the efficient extraction of error syndromes using an ancilla with inferior coherence properties still allows for substantial gains in lifetime. Moreover, the cat code surpasses the decay of the uncorrected cat code by a factor of about 2.2, demonstrating that applying error correction to the logical encoding makes up for the faster error rates introduced by the hardware overhead. The palpable difference in

Uncorrectedcat code

c Decode

0 errors

2 errors

1 error

d Correctb Track error syndromea Encode

Legend

0

0

11

1Initial orientation

–1

+1

P

+Z

+Y+X

+Z

z

+Y+X

–1.7 Re(E) +1.7

–1.7

Im

(E)

+1.

7

E tw tw

tw

Kerr correction

Wait

Measurement correction

tw

TM

TM

TM

TK

TK

TK

TMTM TK

ttot = 13.8 μs ttot = 27.6 μsttot = 0 μs ttot = 27.7 μs ttot = 28.2 μs

Uinit Ufin

2

tj =12ttot

D

D

D

Wigner snapshot

Encode/decode

E D

Error?

0 (no)

1 (yes)

Parity measurement

π pulse on ancilla

Record jump time

ππ

e Process tomography

1

0

F = 0.84

0 0

0

0

0

1

1

1

1

R0

zRtj =34ttot

tj =34ttot

1

0

tr(X1 Xπ/2) = 0.77

1 erroruncorrected

tr(X0M M MX0) = 0.89

1

I

I

XXY Y

ZZ

0

0 errors

Occurrence: 70.4% Occurrence: 25.5% Occurrence: 4.1% 100% of data

1

0

tr(X2 Xπ) = 0.44

2 errorsuncorrected

All errorscorrected

0

arg

−+

+

+

+

−+

±

π

zR π

– π

– π/2

+ π/2

+ π

ππ

Figure 2 | Example of a two-step quantum trajectory executed by the QEC state machine. a, Six cardinal points on the Bloch sphere ρinit are encoded (‘E’) from the ancilla onto even-parity resonator states; green markers indicate the initial coordinate-system orientation. A ‘Wigner [tomography] snapshot’ is shown for ( − )→ ( − )α α

+ +C C0 1 i12

12

; =n 30 ; β is the amplitude of varying coherent displacements ( ˆ β=βD 0 f )

and the average parity ˆ ˆ ˆ†⟨ ⟩ = ⟨ ⟩β βP D PD , where ˆ †= πP ei a a. b, A state machine

for adaptive parity monitoring with delays tw ≈  13 µ s between each measurement. ‘Parity measurement’ rectangles show the Ramsey sequence that maps even [+ /− ] (odd [+ /+ ]) parity onto ancilla | g⟩ (| e⟩ ); the + or − specifies the sign of the π /2 pulse. Diamonds indicate branching on ancilla measurement (0 →  no error, | g⟩ ; 1 →  error, | e⟩ ); ‘π pulse on ancilla’ rectangle indicates ancilla reset (| e⟩  →  | g⟩ ); clocks indicate recording of the error time tj. Dashed purple arrows emphasize the phase difference between 10 and 01 due to θK. Rotations θM are due to cross-Kerr interactions between the readout and storage resonators during ancilla

measurements. The parity (Wigner tomogram origin) matches the best estimate (border colour); tomograms match the expected resonator state as seen in simulations. c, The feedback aligns all states by changing the phase of subsequent resonator drives (for example, for Wigner tomography or decoding pulses) to account for θK and θM. Ancilla tomography after decoding shows the expected rotations of π /2 per error about Z (green markers). Different decoding pulses (‘D’) are chosen in real-time depending on the parity of the final state. d, The correction to obtain the final state ρfin is made via coordinate system rotations ( ˆRz, where z is the qubit axis defined by Pauli matrix σz) by 0 (for 0 errors), − π /2 (for 1 error) or − π (for 2 errors) in software. e, Process tomography results for j = 0, 1 and 2 errors before correction. Ideal Xjπ/2 process matrices are shown in wire-outlined bars. Experimental data for X j

M are shown in solid bars; the values are complex numbers with the amplitude on the vertical axis and an argument specified by the bar colour. Amplitudes of less than 0.01 are not depicted. Process tomography after correction is shown to the right of the arrow. I, the identity matrix; F, fidelity.

© 2016 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

Extra slide: The quantum LC oscillator

LC+Q/2

−Q/2

H =Q2

2C+

Φ2

2LΦ

Classical energy

Rewrite by defining ω = 1/ LC

H =Q2

2C+

12

Cω2Φ2

Harmonic oscillator with “mass” C, “position” Φ and “momentum” Q

H =Q2

2C+

12

Cω2Φ2 [Φ, Q] = iℏ

Q = charge Φ = magnetic flux