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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
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