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Superconducting Transmon Machines

Superconducting Transmon Machines

Hardware Platforms: Superconducting Machines

• CSC591/ECE592 – • Fall 2020

• Professor Patrick Dreher • Research Professor, Department of Science • Chief Scientist, NC State IBM Q Hub

3 November 2020 Superconducting Transmon Quantum 1 5 November 2020 Patrick Dreher Outline • Introduction – Digital versus Quantum • Conceptual design for superconducting transmon QC hardware platforms • Construction of with electronic circuits • • Josephson junction and nonlinear LC circuits • Transmon design with example IBM chip layout • Working with a superconducting transmon hardware platform • Quantum of Two State Systems - Rabi oscillations • Performance characteristics of superconducting transmon hardware • Construct the basic CNOT gate • Entangled and controlled gates • Appendices • References • Quantum mechanical solution of the 2 level system 3 November 2020 Superconducting Transmon Quantum Computers 2 5 November 2020 Patrick Dreher Digital Computation Hardware Platforms Based

on a Base2 Mathematics

• Design principle for a digital computer is built on base2 mathematics

• Identify voltage changes in electrical circuits that map base2 math into a “zero” or “one” corresponding to an on/off state

• Build electrical circuits from simple on/off states that apply these basic rules to construct digital computers

3 November 2020 Superconducting Transmon Quantum Computers 3 5 November 2020 Patrick Dreher From Previous Lectures

It was shown that basic digital electronic gates (ex. NAND etc.) are inconsistent as design primitives for constructing a quantum computer

So if it is known that standard NAND digital logic circuits can’t reproduce the proper quantum mechanical behavior as summarized in the axioms of how is a gate based quantum designed and constructed?

3 November 2020 Superconducting Transmon Quantum Computers 4 5 November 2020 Patrick Dreher Question:

How can one construct a physical piece of hardware that operates in a manner that emulates quantum mechanical behavior as described by the axioms of quantum mechanics?

3 November 2020 Superconducting Transmon Quantum Computers 5 5 November 2020 Patrick Dreher Conceptual Design Requirements for a Gate-based Superconducting Transmon Quantum Computer

• The hardware design must incorporate the ability to o Construct a physical system with (at least) two uniquely addressable quantum mechanical states o Initialize the state of those qubits o Implement arbitrary rotations of the ’s state on the o Construct a set of universal gates o Entangle two qubits o Design a hardware platform that operates with . Decoherence times sufficiently long to implement a circuit with enough depth to accomplish a calculation . A way to measure the final output state of the qubits

Superconducting Transmon Quantum 3 November 2020 6 5 November 2020 Computers Patrick Dreher Consider and LC Resonant Circuit • In classical linear circuit theory, the natural solution for the current in an LC circuit is

i(t) = I0 cosω0t, ω0 =1/ LC

• The current can have any amplitude, independent of the frequency • Energy is stored alternately in the electric field of the capacitor and the magnetic field of the inductor and can have any value at

room temperature U 0  I 2 • Circuit creates an oscillatory behavior

3 November 2020 Superconducting Transmon Quantum Computers 7 5 November 2020 Patrick Dreher Analogy of LC Circuit at Temperature to Harmonic Oscillator Energy Levels

• If the LC circuit is cooled to a superconducting temperature (less than 4o K) then the circuit will begin to display equally spaced quantized energy levels

1 U = ω  n +   0    2 

• This construct provides discrete states

3 November 2020 Superconducting Transmon Quantum Computers 8 5 November 2020 Patrick Dreher Propose Assigning Qubits to LC Resonant Circuit Energy Levels

3 November 2020 Superconducting Transmon Quantum Computers 9 5 November 2020 Patrick Dreher Mapping Qubits to LC Circuit Energy Levels

• Difficulty is that all of the discrete states are equidistant in energy (i.e. energy difference between 0 state and the 1 state is the same as energy difference between all other states n,n+1)

• Impossible to distinguish a |0> state from a |1> state

• Need to construct a mechanism to be able to make this distinction

3 November 2020 Superconducting Transmon Quantum Computers 10 5 November 2020 Patrick Dreher Potential Solution

• If there was some way to set the levels such that their values depended on the magnitude of the current or voltage, then the energy levels would no longer be equally spaced!

3 November 2020 Superconducting Transmon Quantum Computers 11 5 November 2020 Patrick Dreher Nonlinear LC Circuits

• If a nonlinearity could be introduced into the harmonic oscillator energy levels in a way that the energy difference between n=0 and n=1 was different from the energy difference from all other states, it will be possible to selectively address each Graphic: http://hyperphysics.phy- particular transition by tuning the frequency of the astr.gsu.edu/hbase/quantum/hosc.html applied excitation

3 November 2020 Superconducting Transmon Quantum Computers 12 5 November 2020 Patrick Dreher Graphic: Clarke & Wilhelm Comparison Harmonic versus Anharmonic Oscillator

3 November 2020 Superconducting Transmon Quantum Computers 13 5 November 2020 Patrick Dreher What Types of Materials Work at these Extreme Low Temperatures • At these low temperatures, metals such as aluminum (Al) and Niobium (Nb) become superconductors – At low temperatures, an attractive force appears between – When this force gets sufficiently strong compared to thermal vibrations, electrons bind together into “Cooper pairs” with 1 and charge 2q – Cooper pairs form a macroscopic enabling charge to move without scattering or loss, resulting in superconductivity • Superconductivity allows for – extremely low-loss RF transmission lines – Construction of nonlinear inductors using a Josephson junction

3 November 2020 Superconducting Transmon Quantum Computers 14 5 November 2020 Patrick Dreher At Low Tempertures Introduce Superconducting Materials such as a Josephson Tunnel Junction iϕ Circuit Symbols iϕ1 = e 2 ψ1 = ψ 1 e ψ 2 ψ 2

Graphic: Clarke & Wilhelm

• Two superconductors separated by a thin insulating layer • Wave functions for superconducting Cooper pairs decay exponentially in the insulating layer • If the layer is thin enough to allow appreciable tunneling, then phases are no longer independent but are related to each other through the size of the tunneling current Graphic: Ph.D. thesis: Vratislav Michal 3 November 2020 Superconducting Transmon Quantum Computers 15 5 November 2020 Patrick Dreher Josephson Junction as a Nonlinear Inductor

ϕ = ϕ2 − ϕ1 • Effective inductance depends on the current

I = Ic sin ϕ • Looks like a non-linear inductor: origin of Φ dϕ h V = 0 , Φ = is the flux quantum anharmonicity: spacing between energy levels 2π dt 0 2q is not the same dI = I cosϕ dϕ dt c dt 2πV = I ccosϕ Φ0 ⇒ Φ dI – Enables the individual addressing of a single pair of V = 0 2π Ic cosϕ dt states Φ dI – This is in contrast for a linear circuit where all = 0 2 dt states are equally spaced 2πIc 1− sin ϕ Φ dI dI 0 = (I) = L eff 2 dt dt 2πIc 1− (I / Ic) 3 November 2020 Superconducting Transmon Quantum Computers 16 5 November 2020 Patrick Dreher Various Designs for the Non-Linear Elements • Important to know the frequencies that are driving each of the qubits

• Fine tune performance properties of the interactions between qubits

• One can choose either a fixed frequency transmon or a flux tunable transmon to address each of the energy levels

• The IBM quantum computing hardware designers selected the fixed frequency option

• Other designs (such a Rigetti’s gate based quantum computing) hardware platform use a flux tunable transmon 3 November 2020 Superconducting Transmon Quantum Computers 17 5 November 2020 Patrick Dreher Fixed and Tunable Transmon Designs

- Advantages -

• Fixed Frequency Transmon • Flux Tunable Transmon

• Stability (remains calibrated for • Flexibility for faster and/or hours) simpler 2 qubit gates • No flux noise  high • Relaxed junction fabrication that implies avoidance of frequency collisions

3 November 2020 Superconducting Transmon Quantum Computers 18 5 November 2020 Patrick Dreher Fixed and Tunable Transmon Designs

- Disadvantages -

• Fixed Frequency Transmon • Flux Tunable Transmon

• Restricted gates requiring all • Flux noise leads to lower coherence microwave electronics • Flux sensitivity characterized by • Frequency collisions unstable calibrations

3 November 2020 Superconducting Transmon Quantum Computers 19 5 November 2020 Patrick Dreher LC Electrical Properties at Low Temperatures

• Electrical circuits operate at RF rather than optical frequencies • of thermodynamics says that any physical mass at finite temperature will emit electromagnetic radiation that depends on its temperature (Black body radiation) • Want to design the system to be as quiescent as possible from a thermal radiation perspective – that translates into building system at low temperatures

3 November 2020 Superconducting Transmon Quantum Computers Graphic: Wikipedia 20 5 November 2020 Patrick Dreher Low Temperature Operational Requirement

– Want the energy difference between qubit states to be large compared to thermal radiation – Highest frequency for widespread, economical design of instrumentation is ~ 6 GHz – Using equation kT=hυ frequency of 6 GHz corresponds to approximately a temperature of 0.29K – Therefore the operating temperature of a QC superconducting transmon must be much less than 0.3K! – 100x colder than outer space (4 degrees Kelvin)

3 November 2020 Superconducting Transmon Quantum Computers 21 5 November 2020 Patrick Dreher The IBM Superconducting Transmon Design*

• IBM Q systems operate at a temperature of about 15 mK • This low temperature is achieved by using dilution refrigerators

* Image from http://www.wikiwand.com/en/Dilution refrigerator 3 November 2020 Superconducting Transmon Quantum Computers 22 5 November 2020 Patrick Dreher 3 November 2020 Superconducting Transmon Quantum Computers 23 5 November 2020 Slide credit: DougPatrick McClure, Dreher IBM Low Temperature Experimental Apparatus IBM Q Quantum Computer Cryostat

3 November 2020 Superconducting Transmon Quantum Computers 24 5 November 2020 Patrick Dreher 3 November 2020 Superconducting Transmon Quantum Computers 25 5 November 2020 Slide credit: DougPatrick McClure, Dreher IBM IBM T.J. Watson Research Center

CIRCUIT (cQED) • Theory: Blais et al., Phys. Rev. A 69, 062320 (2004) out

• Superconducting transmission line “cavity”

10 µm Superconducting artificial Microwave “atom” in Figure from Introduction to Superconducting Qubits and Quantum Experience Hanhee Paik (2016) 3 November 2020 26 5 November 2020 Superconducting Transmon Quantum Computers Patrick Dreher Superconducting Qubits on a Substrate

3 November 2020 Superconducting Transmon Quantum Computers 27 5 November 2020 Patrick Dreher Transmon Design With Example IBM Chip Layout Blais, et al • Co-planar microstrip formed by gaps in center conductor • Important to properly choose resonator frequency with respect to transmon frequency • Control is achieved by injecting an RF signal from one end • Readout is achieved by looking at either the transmitted or reflected signal

3 November 2020 Superconducting Transmon Quantum Computers 28 5 November 2020 Patrick Dreher(credit IBM research) Working With A Hardware Platform Based on Superconducting Transmon Qubits

3 November 2020 Superconducting Transmon Quantum Computers 29 5 November 2020 Patrick Dreher Two State Quantum Mechanical Systems in a Superconducting Transmon Hardware Platform

3 November 2020 Superconducting Transmon Quantum Computers 30 5 November 2020 Patrick Dreher Importance of Understanding the Device Characteristics of Qubits • An essential aspect for using superconducting transmon qubits is to o Understand their specs through their frequency characteristics and properties o Manually control and manipulate qubit frequencies in order to . Design custom gates . Apply noise reduction methods to extend qubit coherence lifetimes • Need to be able to measure the frequency and the amplitude needed to achieve a π rotation from |0> to |1> • This is essential to perform measurement chracteristics of the qubit, pulse shaping and gate customization

3 November 2020 Superconducting Transmon Quantum Computers 31 5 November 2020 Patrick Dreher Goal for the Studying the Two State Quantum Mechanics Calculation • In order to measure these quantities need to perform some calculations viewing the qubit as a two state quantum mechanical system • Calculate the energy difference between the two states • Solution indicates that the system will oscillate between the two states at the frequency corresponding to the energy difference • From this observation can measure the π-pulse amplitude needed to transition the qubit between state |0> and |1> (calibrated amplitude)

3 November 2020 Superconducting Transmon Quantum Computers 32 5 November 2020 Patrick Dreher Schrodinger Equation ∂ i∂t ψ = H ψ , ∂ ≡ t ∂t • H is an expression for the total energy of the system (kinetic + potential) and is called the Hamiltonian ∂ • For two orthogonal states: i t 1 = H0 1 = E1 1

i∂t 2 = H0 2 = E2 2 • What if we introduce a perturbation that weakly couples states 1 & 2? i∂t 1 = H0 1 +V12 2

i∂t 2 = H0 2 + V21 1 Schrodinger Equation for 2x2 coupled system

i∂=t 112 HV0 + 12

i∂=t 2 HV0 21 + 21 • This can be written

1EV1 00  112   11  iH∂=t   +   =  200EV2  221   22  11 H=+( EE)σ +−( EE) σσ + V, assuming V= V 221 2 0 1 2 zx 12 21 10  1 0   01 σσ0 = , zx=  , σ=  01  0− 1   10 Formulations of Quantum Mechanics

• “Schrodinger Picture:” all time dependence is in the or state vector • “:” all time dependence is in the Hamiltonian • “:” hybrid in which some time dependence is in both the operator and the state vector – Particularly useful when considering a small perturbation to a solved system: express solution to perturbed system in terms of solutions to the unperturbed system • See the Appendix for the detailed quantum mechanical calculation Quantum Mechanics of Rabi Oscillations

• When a two-level system is coupled to a driving field at precisely the frequency corresponding to the energy difference between the states, the system will oscillate between the two states at the Rabi frequency | >= ( )|0 > + ( • Initial conditions (0) = 1 𝜓𝜓 𝑐𝑐𝑔𝑔 𝑡𝑡 𝑐𝑐𝑒𝑒 𝑡𝑡� (0) = 0 𝑐𝑐𝑔𝑔 𝑒𝑒 𝑐𝑐 2 1 Pg (t) = cg (t) = (1+ cos Ω t) 2 R ΩR t 2 1 P (t) = c (t) = (1−cos Ω t ) e e 2 R 2 Ω = 1 H 0 R  I Example: Creating Gates with Rabi Oscillations

3 November 2020 Superconducting Transmon Quantum Computers 37 5 November 2020 Patrick Dreher Creating Rotations: Rabi Oscillations

• Suppose a time-varying field is applied along the x direction

• The Hamiltonian becomes

• Move to a frame rotating at

• Note that rotating a state vector by φ is equivalent to rotating the coordinates by -φ, so the state vector in the rotation frame is

3 November 2020 Superconducting Transmon Quantum Computers 38 5 November 2020 Patrick Dreher Hamiltonian in the Rotating Frame

3 November 2020 Superconducting Transmon Quantum Computers 39 5 November 2020 Patrick Dreher Rotating Wave Approximation

• Putting the pieces together:

• Neglect the frequency components near twice ω0 since they will tend to average out: Rotating Wave Approximation

∆ = ωd - ω0

3 November 2020 Superconducting Transmon Quantum Computers 40 5 November 2020 Patrick Dreher of State Vector in Rotating Frame

• If the drive frequency matches the qubit frequency, ∆ = 0.

3 November 2020 Superconducting Transmon Quantum Computers 41 5 November 2020 Patrick Dreher Example: Rotation About x Axis

1

P (t) 0 0.9 P (t) 1 0.8

0.7

0.6

0.5 • Note that the rate of rotation can be P controlled by varying either the 0.4 0.3 amplitude or the phase of the 0.2

microwave drive signal 0.1

0 0 2 4 6 8 10 12 14 t 3 November 2020 Superconducting Transmon Quantum Computers 42 5 November 2020 Patrick Dreher Gates/Rabi Oscillations/Noise Mitigation

This is a key point for an important • Recall that when driven at resonance error mitigation technique: Richardson • “π-pulse”: ΩRt = π inverts the state extrapolation

• “π/2-pulse”: ΩRt = π / 2 creates equal superposition of states

• Key point: you can flip a state or create a superposition state by 0 controlling the pulse length & controlling the phase of the z 1 (0 − 1 ) excitation 2 1 θ ( 0 − i 1 ) • X gate: pi-pulse with φ = −π /2 2 y φ′ 1 • Y gate: pi-pulse with φ = π 1 x (0 + i 1 ) (0 + 1 ) 2 • Hadamard gate: pi/2-pulse with φ = −π /2 2 1 • Z gate: Rotations around the z axis can be accomplished simply by resetting the phase reference of the RF drive! Result: zero time and zero error (Virtual Z Gate) Example: CNOT Gate

3 November 2020 Superconducting Transmon Quantum Computers 44 5 November 2020 Patrick Dreher Constructing a CNOT Gate With a Superconducting Transmon Hardware Platform • Demonstrate can construct CNOT gate • Consider Hamiltonian coupling two qubits where 1 and 2 refer to qubits 1 and 2 and ZZ refers to the ZZ interaction

= , , 4𝑍𝑍𝑍𝑍 𝑍𝑍𝑍𝑍 𝐸𝐸1 2 1 2 𝐻𝐻1 2 𝜎𝜎𝑍𝑍 ⊗ 𝜎𝜎𝑍𝑍 • This interaction term corresponds to a unitary time-evolution given by

• Turn interaction on and off over a time = ℎ , 𝑍𝑍𝑍𝑍 3 November 2020 Superconducting Transmon Quantum𝑡𝑡 Computers2𝐸𝐸1 2 45 5 November 2020 Patrick Dreher • Use computational basis |0,0> with a corresponding vector of (1,0,0,0) to construct the unitary operation

1 0 0 0 0 0 0 = exp[ ] , 4 0 0 0 𝑍𝑍𝑍𝑍 𝑖𝑖𝑖𝑖 −𝑖𝑖 𝑈𝑈1 2 0 0 0 1 • This unitary operation can be combined− with𝑖𝑖 rotations around z of 1 2 each qubit Rz (-π/2) and Rz (π/2) up to a global phase factor

3 November 2020 Superconducting Transmon Quantum Computers 46 5 November 2020 Patrick Dreher C-Phase Gate

Control qubit

11

Target qubit

1 0 0 0 0 1 0 0 = [ ( ) ( )] = exp[ ] , 2 2 , 4 0 0 1 0 𝑖𝑖 𝑗𝑗 𝑍𝑍𝑍𝑍 1 1 𝑍𝑍 1 2 𝑖𝑖𝑖𝑖 0 0 0 1 𝑐𝑐𝑈𝑈 𝑅𝑅 −𝜋𝜋� ⊗ 𝑅𝑅𝑍𝑍 −𝜋𝜋� 𝑈𝑈 −

3 November 2020 Superconducting Transmon Quantum Computers 47 5 November 2020 Patrick Dreher CNOT Gate Relation to C-Phase Gate

• This cU11 c-Phase gate corresponds to a phase shift of π on the target qubit when the control qubit is in the excited state |1> • The CNOT and C-Phase gates differ by only single qubit rotations • The CNOT gate can be constructed using the C-Phase Gate with Hadamard gates on the target qubit Control qubit

= 11 Target qubit H H

3 November 2020 Superconducting Transmon Quantum Computers 48 5 November 2020 Patrick Dreher CNOT Gate Calculation

• Write the outer product representation of the combined Hadamard and c-Phase gates

1 1 0 0 1 0 0 0 1 1 0 0 1 1 1 0 0 0 1 0 0 1 1 1 0 0 = exp( ) 4 2 0 0 1 1 0 0 1 0 2 0 0 1 1 𝑖𝑖𝑖𝑖 − − 𝑈𝑈𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 0 0 1 1 0 0 0 1 0 0 1 1

− 1 0 0− 0 − 0 1 0 0 = exp( ) 4 0 0 0 1 𝑖𝑖𝑖𝑖 𝑈𝑈𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 0 0 1 0

3 November 2020 Superconducting Transmon Quantum Computers 49 5 November 2020 Patrick Dreher Performance Metrics for Superconducting Transmon Hardware

3 November 2020 Superconducting Transmon Quantum Computers 50 5 November 2020 Patrick Dreher T1 – Relaxation Time • Initialize with qubit in the |0> • Put the qubit into the |1> state by applying an X-gate (π-pulse) • Wait a specified period of time and then measure in the relaxation time of the qubit from |1> state  |0> • Find the relaxation rate by fitting and exponential decay curve to the data

IBM figure from Device Characteristics presentation

3 November 2020 Superconducting Transmon Quantum Computers 51 5 November 2020 Patrick Dreher Qubit De-Phasing (T )

• Initialize with qubit in the ground state𝜙𝜙 |0> • Transform the qubit into a superposition state • Allow the qubit state to evolve over time • Measure the qubit state (dephasing)

3 November 2020 Superconducting Transmon Quantum Computers 52 5 November 2020 Patrick Dreher Overall Qubit Decoherence Measurements and Characteristics

• Measure a loss of due to interactions with environmental factors

• T1 is a relaxation time • T is a dephasing time • T is the overall decoherence time 𝜙𝜙2 = + 1 1 1 𝑇𝑇2 2𝑇𝑇1 𝑇𝑇𝜙𝜙 IBM figure from Device Characteristics presentation 3 November 2020 Superconducting Transmon Quantum Computers 53 5 November 2020 Patrick Dreher Specification Data for an IBM Q Hardware Platform

Single-qubit Qubit T1 (µs) T2 (µs) Frequency (GHz) Readout error U2 error rate CNOT error rate Date Q0 58.8833343 65.40557317 4.968658551 1.12E-02 2.63E-04 cx0_1: 7.066e-3 Tue Oct 27 2020 03:06:12 GMT-0400 (EDT) Q1 155.7258209 59.31927948 4.770375545 4.17E-02 2.69E-04 cx1_0: 7.066e-3, cx1_2: 1.823e-2 Tue Oct 27 2020 03:06:12 GMT-0400 (EDT) Q2 81.77066486 98.2056677 5.015085378 5.55E-02 3.38E-04 cx2_1: 1.823e-2, cx2_3: 1.060e-2 Tue Oct 27 2020 03:06:12 GMT-0400 (EDT) Q3 32.06346406 25.34362315 5.259407406 2.39E-02 6.52E-04 cx3_2: 1.060e-2, cx3_4: 1.292e-2 Tue Oct 27 2020 03:06:12 GMT-0400 (EDT) Q4 61.77515414 120.4829221 4.997495545 8.70E-03 2.85E-04 cx4_3: 1.292e-2 Tue Oct 27 2020 03:06:12 GMT-0400 (EDT)

3 November 2020 Superconducting Transmon Quantum Computers 54 5 November 2020 Patrick Dreher Entangled Qubits

3 November 2020 Superconducting Transmon Quantum Computers 55 5 November 2020 Patrick Dreher Multi-Qubit

• One of the advantages of quantum computing is the ability to entangle qubits

• This expands the computational power of these hardware platforms

• The process of entangling qubits is connected to o Understanding the operational frequency of each qubit o Ability to shape and customize the pulses for each calculation o Ability to successfully read-out the final states of the qubits

3 November 2020 Superconducting Transmon Quantum Computers 56 5 November 2020 Patrick Dreher Hamiltonian for Two Coupled Qubits

• Hamiltonian for two coupled qubits, each with an RF drive perturbation 1 1 1 H = ωσ z +V cos ω rf t +φ σ x + ω σ z +V cos ω rf t +φ σ x + ω σ x ⊗σ x  1 1 1 ( 1 1 ) 1  2 2 2 ( 2 2 ) 2  xx 1 2 2 2 2

• If V2=0, and we drive the first qubit at the resonance frequency of the rf second, we have the cross-resonance condition ω1 = ω2 • After multiple transformations and making the rotating wave approximation, this can be expressed 1 V ω eff = ωeff z x z y eff 1 xx H  xx (cosφ1σ1 ⊗σ 2 + sinφ1σ1 ⊗σ 2 ), ωxx = 2 2(ω2 − ω1 )

eff • When ωxx t = π / 2 this implements a ZX gate, and along with two single qubit gates this enables the realization of a CNOT

3 November 2020 Superconducting Transmon Quantum Computers 57 5 November 2020 Patrick Dreher Cross-resonant Coupling

• Frequency switches on coupling • Amplitude controls the gate speed • Phase determines the two-qubit gate

3 November 2020 Superconducting Transmon Quantum Computers 58 5 November 2020 Patrick Dreher Rigetti and Devoret Entangling two qubits with a

• “Dispersive coupling”: resonator detuned from either qubit frequency

3 November 2020 Superconducting Transmon Quantum Computers 59 5 November 2020 Patrick Dreher Graphic: Chow PhD thesis Interactions between Qubits

• Co-planar microstrip resonator formed by gaps in center conductor

• Important to properly choose resonator frequency with respect to transmon frequency

• Control is achieved by injecting an RF signal from one end Blais, et al

• Readout is achieved by looking at either the transmitted or reflected signal

3 November 2020 Superconducting Transmon Quantum Computers 60 5 November 2020 Patrick Dreher Control and Read-out

• When the microstrip resonator is detuned from the transmon frequency, interaction withthe transmon splits the resonator response into two modes, depending on the transmonstate

• Sending in a pulse near ωr enables you to read-out the state either from the phase, or the amplitude • Sending in a pulse detuned from the microstrip resonator but tuned to the qubit frequency rotates the state, but does not make a measurement (there is no information about the state in the reflected signal, the resonator is so far off resonance)

3 November 2020 Superconducting Transmon Quantum Computers 61 5 November 2020 Blais, etPatrickal Dreher Final Summary and Observations

3 November 2020 Superconducting Transmon Quantum Computers 62 5 November 2020 Patrick Dreher Development Trajectory for Superconducting Transmon Quantum Computing Hardware Platforms

• After exploring several different designs for qubit placement on a substrate, the IBM designers have committed to expanding a “heavy hex” topology

• Tradeoffs IBM’s o less crosstalk with fewer qubit Manhattan interconnects QC platform o Because qubits must be adjacent 65 qubits for computation, it will require more SWAP gates to be inserted into the circuits o Optimization with “pulse control” 3 November 2020 Superconducting Transmon Quantum Computers 63 5 November 2020 Patrick Dreher Early Digital Computing Hardware ENIAC Machine Room ~ 1950

3 November 2020 Superconducting Transmon Quantum Computers 64 5 November 2020 Patrick Dreher Digital Computer Technology Seventy Years Later in 2020

Google’s London data center SUMMIT for Cloud Computing Oak Ridge National Laboratory 3 November 2020 Superconducting Transmon Quantum Computers 65 5 November 2020 Patrick Dreher Quantum Computing Technology IBM Q Machine Room - 2020

3 November 2020 Superconducting Transmon Quantum Computers 66 5 November 2020 Patrick Dreher We are in the Early Days of Quantum Computing

Source: Anthony Annunziata, IBM IBM Q Quantum Computing Machine Room, 2020 3 November 2020 Superconducting Transmon Quantum Computers 67 5 November 2020 Patrick Dreher Quantum Computing Technology in 2090

3 November 2020 Superconducting Transmon Quantum Computers 68 5 November 2020 Patrick Dreher Questions

3 November 2020 Superconducting Transmon Quantum Computers 69 5 November 2020 Patrick Dreher Appendices

• References • Quantum mechanical solution of the 2 level system*

*Some of the slides are shared as a courtesy from a lecture by Professor Stancil to the CSC591 course in 2018

3 November 2020 Superconducting Transmon Quantum Computers 70 5 November 2020 Patrick Dreher References

From D. Stancil, Quantum Computing CSC591/ECE592, 24 Sept. 2018

• A. Blais, et al, “Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation,” Phys. Rev. A, Vol. 9, 062320 (2004) • J. Koch, et al, “Charge-insensitive qubit design derived from the Cooper Pair Box,” Phys. Rev. A Vol. 76, 042319 (2007). • J. Clarke & F. Wilhelm, “Superconducting quantum ,” Nature, Vol. 453, 19 June 2008, p. 1031. • J.M. Chow, “Quantum Information Processing with Superconducting Qubits,” Ph.D. Dissertation, Yale Univ., 2010. • C.T. Rigetti, “Quantum Gates for Superconducting Qubits,” Ph.D. Dissertation, Yale Univ., 2009. • C. Rigetti and M. Devoret, “Fully microwave-tunable universal gates in superconducting qubits with linear couplings and fixed transition frequencies,” Phys. Rev. B vol. 81, 134507 (2010) • T. Van Duzer and C.W. Turner, “Principles of Superconductive Devices and Circuits,” Elsevier, 1981. • A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J.M. Chow and J.M. Gambetta, “Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets,” Nature, Vol. 549, p. 242 (2017).

3 November 2020 Superconducting Transmon Quantum Computers 71 5 November 2020 Patrick Dreher Reference

A Quantum Engineer’s Guide to Superconducting Qubits

Krantz, Kjaergaard, Yan, Orlando, Gustavsson, and Oliver

arXiv 1904.06560

3 November 2020 Superconducting Transmon Quantum Computers 72 5 November 2020 Patrick Dreher Equations of in the Interaction Picture

• Let us express the Hamiltonian as a sum of two terms, the larger of which is time independent and for which solutions are known, and a smaller

perturbation that contains all of the time dependence: H = H0 + V (t)

• In the Schrodinger picture: i∂t ψ (t) = H ψ (t) S S • The transformation for the state vector into the Interaction picture: ψ (t) = eiH0t/ ψ (t) I S • Equation of motion:

iH 0t / i ∂ ψ (t) = i ∂ e iH0t/  t  t ψ(t) i ∂ ψ(t) = e V ψ (t) I ( S )  t I S iH t / = e 0 (i∂ − H )ψ(t) = eiH0t/Ve−iH0t/ eiH0t/ ψ (t) t 0 S   S iH t/ VI (t ) 0  ψ (t ) = e (H + V− H ψ(t) I 0 0 ) S

i ∂t ψ (t) = VI (t)ψ (t) • Only depends on time-dependent perturbation:  I I Solution to Equation of Motion

• Construct an interaction picture solution by adding up eigenstates of the unperturbed Hamiltonian but with time-dependent coefficients: ψ = (t) I ∑ cn (t) n • Substitute into equation of motion: n = iH0t/ −iH0t/ i∂t ∑c n n e Ve ∑c n n n n iH t / −iE t / i c 0  n  ∑ n n = e V ∑cne n n n

iH0 t / −iE nt / i c n m n = m e V c e n ∑   ∑ n n  n δmn meiEmt /

eiωmnt c , icm = ∑Vmn n n

Vmn = m V n , ωmn = (Em − En )/  = −ωnm Application to 2-Level System

1 1 H = (E + E )σ + (E − E )σ +Vσ • Recall: 2 1 2 0 2 1 2 z x • Choose zero of energy as the average, and V caused by sinusoidal RF signal: 1 H = − ω σ +V cos ω rf t +φ σ  21 z 0 ( ) x 2

eiωmntc • icn = ∑Vmn n become: n

ω i c = V cos ω rf t +φ ei 12t c 1 0 ( ) 2 ω i c = V cos ω rf i 21t  2 0 ( t +φ)e c1

ω rf+φ −i ω trf+φ V0  i( t ) ( )  −iω t ic = e + e e 21c 1 2   2

ω rf+φ ω rf+φ V0  i( t ) i ( t )  iω t ic = e + e e 21 c 2 2   1 Solution in the Rotating Wave Approximation

• Assume the energy associated with the RF frequency is close to the transition energy between the two states: rf ωω=21 +∆, ∆  ω21 • Then keep only the low-frequency components to the solution (Rotating

rf Wave Approximation): VVi(ωω−+21 ) ti φ iti∆+φ ic = 00 e c= e c 122 22

rf VV00−−−i(ωω21 ) ti φ −∆−itiφ ic2= e c 11= e c • Substituting the first equation 22into the second gives a second-order Diff. Equation for c : 2 1 V0 c11−∆ ic  + c 1 =0 2 • If the system starts off in state 1, then it is easily verified that the solution is

2 i∆/2 c( t )= Ae cos( Ω t / 2) 2 V0 1 Ω= ∆ +  2A −∆ii/2 −φ ∆Ω ct()= i e icosΩ− t /2 sin Ωt /2 − 2 ( ) ( ) 22 2 1/2 V0 22 + = ⇒ = +∆ cc121 A( 1(  / V0) ) Interaction Picture Calculation Driving at Resonance: Rabi Oscillations • Calculate the 2 states RF driving frequency (See Appendix) • If the RF driving frequency corresponds to the energy difference between the two states, then rf ∆=0, ωω = 21 • The coefficients then become ct1( )= cos( ΩR t / 2) −iφ c2 ( t )=−Ω ie sin( R t / 2)

Ω=R V0 /  • Let state 1 correspond to the ground state, and state 2 to the excited state. The of finding the system in each state is given by

2 2 1 ctgR()=( cos( Ω t /2)) =( 1cos +Ω Rt) 2 2 2 1 cteR()=( sin( Ω t /2)) =( 1cos −Ω Rt) 2

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