quantum control 量子控制 - sustech · 2021. 4. 28. · quantum control 量子控制 —the...

Quantum Control 量子控制— The theory of controlling qubits and the

implementations on superconducting quantum

processors

Xiu-Hao Deng 邓修豪 (dengxh@sustech.edu.cn)Shenzhen Institute of Quantum Science and Engineering (SIQSE)

& Guangdong Provincial Key Laboratory of Quantum Science and Engineering

https://faculty.sustech.edu.cn/dengxh

What is Quantum Control ?

Control of physical systems whose

behavior is dominated by the laws of

quantum mechanics.

——2003: Jonathan Dowling and Gerard Milburn,

“The development of the general principles

of quantum control theory is an essentialtask for a future quantum technology.”

Classical

Control--

Cybernetic

Engineering控制论

Quantum Control -- A computer architecture

科学+工程

E.g. NISQ 量子计算机

Xiang Fu, IEEE Micro, 2018

Quantum Control -- A quantum system

Final state readout

Control field

Initial stateQuantum evolution

Quantum Control -- A quantum system

Electric fieldMagnetic fieldEM fields: Laser, MicrowaveHeatMechanical force… SC qubit: Microwave

Quantum Control – Quantum Dynamics

https://faculty.sustech.edu.cn/dengxh

Credit to Q. Guo @ Q Dynamics group

Quantum Control – Quantum Dynamics group

https://faculty.sustech.edu.cn/dengxh

Quantum Control – Quantum Dynamics

➢ Schrodinger Equation

➢ Qubit(s) under control

➢ Qubit(s) with noise

➢ Quantum control：Designing the Quantum Dynamics

Qubit Initialization

quantum gates

quantum readout,

etc

➢ Closed Quantum System

• Time evolution of quantum state

• Quantum Control in Close System

• Solving quantum evolution due to time-

dependent Hamiltonian

➢ Open Quantum System

• Observed quantum system coupled to

environment

• Kraus formalism

• Markovian approximation

• Master Equation

To control quantum dynamics?

How does the quantum system evolve with certain Hamiltonian?

What control field should be added to the Hamiltonian to let the system evolve as wish?

What control field should be added to the Hamiltonian to let the subsystem evolve as wish?

What control field should be added to the Hamiltonian to let the partial system in the subspace evolve as wish?

How does the evolution deviate subject to noises?

What control could decouple the target system from noises?

What control could drive the target system’s expected evolution robustly and faithfully?

Many possible control fields could drive the system to be close to the same target evolution, but which one is

optimal, fastest, and/or robust?

➔ Given target evolution (operation), search for appropriate controls

What you should learn from this course

➢Understand the principle behind the driven qubit dynamics

➢Know what theoretical tools you could use for a specific question

➢Know an explicit map of knowledge for quantum control

➢Know some of the frontier problems to resolve

What you do after this course by yourself

➢Derive the equations

➢Homework

➢Solve a specific problem using the introduced tools

Outline

➢Time evolution

• Closed quantum system dynamics

• Geometric picture

• Open quantum system dynamics

➢Quantum control

• 1 QB gates

• 2 QB gates

• Quantum control in open quantum system and dynamical decoupling

• Dynamical correction gates

Time evolution of quantum state

➢ Closed quantum System

Schrödinger equation – a ‘Derivation’:

Evolution (Unitary) operator–

a ‘Integration’:

Dyson series: an interferent summation of all the

possible path.

Feynman Path integral

Geometric picture of single qubit state and operations

➢ Qubit state ➔ Bloch sphere ➢ Qubit operation

Suppose the operation element is

Polar decompose the M

Exercise: programming the representation of

arbitrary qubit state on a Bloch sphere

Geometric picture of single qubit state and operations

• Euler rotation

Ed Barnes, Phys. Rev. Lett. 109, 060401 (2012)XH Deng, In preparation

• Quaternion

J. Phys. A: Math. Theor. 48 (2015) 235302

Quaternion:

Rules:

➢ Parametrization of qubit operations

Time evolution of open qubit system

➢ Open Quantum System

could be non-unitary！

Time evolution of open qubit system

➢ Open Quantum System

Two different ways of describing the evolution of an open system:

1. The evolution of a composite system (including the system of interest and a bath) tracing

over arrived at a description of the open evolution via the operator sum. ➔ Kraus

Operators

2. Quite abstract. And only defining the properties of the linear map (super-operator)

describing the evolution in order to arrive at an acceptable (physical) evolved state (that

still possess the characteristics of a density operator). ➔Master equation

Time evolution of open qubit system

➢ Kraus Operators

The Kraus representation is not unique !

➢ Amplitude-damping channel

The density matrix evolves as

Time evolution of open qubit system

➢ Phase-damping channel

➢ Depolarization channel

➢ Bit flip channel

➢ Bit-Phase flip channel

1. Please plot the transformation of Bloch sphere.

2. Please plot the transformation of Bloch sphere.

3. Please derive the Kraus operator and plot the transformation

of Bloch sphere.

Time evolution of open qubit system

➢ Master equation

Outline

➢Time evolution

• Closed quantum system dynamics

• Geometric picture

• Open quantum system dynamics

➢Quantum control

• 1 QB gates

• 2 QB gates

• Quantum control in open quantum system and dynamical decoupling

• Dynamical correction gates

Single qubit gate

𝜎𝑥

𝐻 =1

2𝜔𝑞𝜎𝑧 + Ω𝑑 cos(𝜔𝑑𝑡 + 𝜙𝑅) 𝜎𝑥

➔ Rotating frame transformation

𝐻𝑟𝑜𝑡 =1

2Δ𝜎𝑧 + Ω𝑥𝜎𝑥 + Ω𝑦𝜎𝑦

= 𝑀 ⋅ Ԧ𝜎Δ = 𝜔𝑞-𝜔𝑑

|1>

|0>

2-level 𝐻𝑑𝑟𝑖𝑣𝑒 =

𝑘

𝜎x(Ξ𝑘𝑒−𝑖𝜔𝑑

𝑘𝑡 + Ξ𝑘

∗𝑒𝑖𝜔𝑑𝑘𝑡)

How to implement Z gate and XY gate?

Problems of 1Q gate in realistic system

𝐻 =1

2𝜔𝑞 + 𝛿𝑧 𝜎𝑧 + Ω𝑑(𝑡) cos 𝜔𝑑𝑡 + 𝜙𝑅 𝜎𝑥 + 𝛿𝑥𝜎𝑥 + 𝛿𝑦𝜎𝑦

• Open system: Noises by environment

NATURE COMMUNICATIONS, 7:11527 (2016)

Quantum Stud.: Math. Found. (2020) 7:23–47

➔ Need for engineering Ω𝑑(𝑡)

𝐻 =1

2𝜔𝑞𝜎𝑧 + Ω𝑑 cos(𝜔𝑑𝑡 + 𝜙𝑅) 𝜎𝑥

Problems of 1Q gate in realistic system

𝐻 =1

2𝜔𝑞 + 𝛿 𝜎𝑧 + Ω𝑑 cos 𝜔𝑑𝑡 + 𝜙𝑅 𝜎𝑥

• In multi qubit system: Spectator, Intruder

Xiu-Hao Deng, arXiv:2103.08169 (2021)

Problems of 1Q gate in realistic system

𝜎𝑥

transmon

Leakage

Gaussian pulse

𝐻𝑑𝑟𝑖𝑣𝑒 = Ω 𝑡 𝑎𝑒−𝑖𝜔𝑡 + Ω′ 𝑡 𝑎†𝑒𝑖𝜔𝑡

𝐻𝑇𝑟𝑎𝑛𝑠𝑚𝑜𝑛 = 𝜔𝑎†𝑎 +𝛼

2𝑎†𝑎†𝑎𝑎 + 𝐻𝑑𝑟𝑖𝑣𝑒

Outline

➢Time evolution

• Closed quantum system dynamics

• Geometric picture

• Open quantum system dynamics

➢Quantum control

• 1 QB gates

• 2 QB gates

• Quantum control in open quantum system and dynamical decoupling

• Dynamical correction gates

Two qubit gate

(Quintana, PRL 110, 173603 (2013))

int

1 0 0 0

0 cos( ) sin( ) 0( )

0 sin( ) cos( ) 0

0 0 0 1

gt i gtU t

i gt gt

− = −

00 10 01 11

Beam-Splitter

• Landau-Zenner

𝑈𝜋

𝑔=

1 00 0

0 0−𝑖 0

0 −𝑖0 0

0 00 1

= 𝑖𝑆𝑊𝐴𝑃

Two qubit gate

• CZ

Phys. Rev. Lett. 125, 240503 (2020)

Two qubit gate

• CZ

Time evolution of the system:

Problems of qubit gate in realistic system

➢ Pulse shaping:

• Pulse optimization: GRAPE, Krotov, CRAB, evolutionary algorithm, etc

• Analytical methods

➢ Close-loop quantum control….

Outline

➢Time evolution

• Closed quantum system dynamics

• Geometric picture

• Open quantum system dynamics

➢Quantum control

• 1 QB gates

• 2 QB gates

• Quantum control in open quantum system and dynamical decoupling

• Dynamical correction gates

Quantum Control in open quantum system

• Use a π pulse to eliminate dephasing due to varying precession frequencies

• Works if noise is slow

Approximate illustration

➢ Dynamical decoupling-Spin Echo

https://www.youtube.com/watch?v=EDyxBWXp6IU

Quantum Control in open quantum system

1. Rotational angle error

2. Rotational axis error

3. Relaxation error

➢ Noises and robust dynamical decoupling

Application of DD in two qubit gate

➢ Noises and robust dynamical decoupling

Noises

arXiv:2104.02669

Outline

➢Time evolution

• Closed quantum system dynamics

• Geometric picture

• Open quantum system dynamics

➢Quantum control

• 1 QB gates

• 2 QB gates

• Quantum control in open quantum system and dynamical decoupling

• Dynamical correction gates

Errors in multi-qubit systems

➢ ZZ interaction: 𝛿 ∝ 𝑔

➢ XX interaction:

• TLS-TLS 𝛿 = 0

• Transmon-TLS 𝛿~𝑔

• Transmon-Transmon 𝛿~𝑔2

Δ

• Transmon-coupler-transmon 𝛿~𝑔4

Δ3

• Unwanted interaction ➔ Stark Effect/AC Stark Effect , Zeeman Effect

t s𝑔

Target qubit Spectating qubit

XH Deng, arXiv:2103.08169

AP

S M

M 2

021

-F3

0.00

10

Errors in multi-qubit systems

• Unwanted interaction ➔ Stark Effect/AC Stark Effect , Zeeman Effect

𝑔

𝑔

𝑔

XH Deng, arXiv:2103.08169

AP

S M

M 2

021

-F3

0.00

10

Errors in multi-qubit systems

• Unwanted interaction ➔ Stark Effect/AC Stark Effect , Zeeman Effect ➔ correlated errors

t s𝑔

Target qubit Spectating qubit

line-splitting invisible

line-splitting significant

XH Deng, arXiv:2103.08169

AP

S M

M 2

021

-F3

0.00

10

Errors in multi-qubit systems

• Unwanted interaction ➔ Stark Effect/AC Stark Effect , Zeeman Effect ➔ Correlated errors

line-splitting invisible → Spectral broadening, Decoherence

line-splitting significant

XH Deng, arXiv:2103.08169

AP

S M

M 2

021

-F3

0.00

10

Errors in multi-qubit systems

• Unwanted interaction ➔ Stark Effect/AC Stark Effect , Zeeman Effect ➔ Correlated errors

line-splitting invisible → Spectral broadening, Decoherence → Spectator

line-splitting significant

Gate error >0.4%

arXiv:2101.01854

PR Applied 14, 024042 (2020)

PR Applied, 12, 054023 (2019)

Nature 574, 505 (2019).

XH Deng, arXiv:2103.08169

AP

S M

M 2

021

-F3

0.00

10

Errors in multi-qubit systems

• Unwanted interaction ➔ Stark Effect/AC Stark Effect , Zeeman Effect ➔ Correlated errors

line-splitting invisible → Spectral broadening, Decoherence → Spectator

line-splitting significant → Control-rotation errors

XH Deng, arXiv:2103.08169

AP

S M

M 2

021

-F3

0.00

10

Errors in multi-qubit systems

• Unwanted interaction ➔ Stark Effect/AC Stark Effect , Zeeman Effect ➔ Correlated errors

line-splitting invisible → Spectral broadening, Decoherence → Spectator

line-splitting significant → Control-rotation errors

+ Inhomogeneous driving

XH Deng, arXiv:2103.08169

AP

S M

M 2

021

-F3

0.00

10

Errors in multi-qubit systems

• Unwanted interaction ➔ Stark Effect/AC Stark Effect , Zeeman Effect ➔ Correlated errors

line-splitting invisible → Spectral broadening, Decoherence → Spectator

line-splitting significant → Control-rotation errors

+ Inhomogeneous driving → Intruder

XH Deng, arXiv:2103.08169

AP

S M

M 2

021

-F3

0.00

10

Targeted-correction gates

XH Deng, arXiv:2103.08169

AP

S M

M 2

021

-F3

0.00

10

Targeted-correcting error

for measured δ and

driving inhomogeneity

• For large 𝛿 due to intruder

Targeted-correction gates

• For large 𝛿 due to intruder(s)

+ GRAB

(Y Song & XH Deng, to be submitted)

Analytical theory giving:

XH Deng, arXiv:2103.08169

AP

S M

M 2

021

-F3

0.00

10

X π-gate fidelity

>0:999fidelity

>0:9999

X π/8-gate

Experimental data are used.

Targeted-correction gates

• For large 𝛿 due to intruder(s)

GRAB >= GRAPE + CRAB + Tensorflow

(Y Song & XH Deng, to be submitted)

GRAB

GRAPE

CRAB

Implementing a single qubit X-rotation in strong

intruded regime

XH Deng, arXiv:2103.08169

AP

S M

M 2

021

-F3

0.00

10

Error-robust gates

• For small 𝛿 due to spectator(s) AP

S M

M 2

021

-F3

0.00

10

Robust to variation of parameters

Error-robust gates

AP

S M

M 2

021

-F3

0.00

10

In perturbative regime:

Error-robust gates

AP

S M

M 2

021

-F3

0.00

10

Error-robust gates

AP

S M

M 2

021

-F3

0.00

10

Error-robust gates

AP

S M

M 2

021

-F3

0.00

10

Error-robust gates

AP

S M

M 2

021

-F3

0.00

10

Error-robust gates

AP

S M

M 2

021

-F3

0.00

10

Error-robust gates

AP

S M

M 2

021

-F3

0.00

10

Error-robust gates

• For small 𝛿 due to spectator(s)

Length

Curvature

Torque

Rotational angle

Noise cancelling conditions:

Scientific reports 5, 1 (2015)NJP20, 033011 (2018)PRA 99, 052321 (2019)

XH Deng, arXiv:2103.08169; YJ Hai & XH Deng, to be submitted

AP

S M

M 2

021

-F3

0.00

10

Error-robust gates

• High fidelity plateau

1st order error cancellation 2nd order error cancellation

XH Deng, arXiv:2103.08169; YJ Hai & XH Deng, to be submitted

AP

S M

M 2

021

-F3

0.00

10

Error-robust gates

XH Deng, arXiv:2103.08169; YJ Hai & XH Deng, to be submitted

AP

S M

M 2

021

-F3

0.00

10

Error-robust gate X gate with Cosine pulse

• Single qubit X gate of two transmon qubits

Experimental data are used.

Error-robust gates

• iSWAP gate of two transmon qubits

XH Deng, arXiv:2103.08169; YJ Hai & XH Deng, to be submitted

Error-robust pulse iSWAP gate with Cosine pulse

AP

S M

M 2

021

-F3

0.00

10

Experimental data are used.

Summary and observation

(YJ Hai & XH Deng, to be submitted)

(Y Song & XH Deng, to be submitted)

(XH Deng et al, arXiv:2103.08169)

+ GRAB

• The intruder regime

• The spectator regime

Observations:

• Fewer pulse parameters for optimization.

• Less control lines and easier calibration?

• Quantum computing with large always-on interactions?

➔ SC qubits

QD spin qubits

• Gate error model

AP

S M

M 2

021

-F3

0.00

10

Outline

➢Time evolution

• Closed quantum system dynamics

• Geometric picture

• Open quantum system dynamics

➢Quantum control

• 1 QB gates

• 2 QB gates

• Quantum control in open quantum system and dynamical decoupling

• Dynamical correction gates