Quantum Computers

ERIK LIND. PROFESSOR IN NANOELECTRONICS Erik Lind EIT, Lund University 1

Quantum Computers

Classical Digital Computers

Why quantum computers?

Basics of QM

• -

• Optical qubits

• Superconducting Qubits

• Topological Qubits Erik Lind EIT, Lund University 2

Classical Computing

• We build digital electronics using CMOS

• Boolean Logic

• Classical Bit – 0/1 (Defined as a voltage level 0/+Vdd)

+vDD +vDD

+vDD

Inverter NAND Logic is built by connecting gates Erik Lind EIT, Lund University 3

Classical Computing

• We build digital electronics using CMOS

• Boolean Logic

• “Classical Bit” – 0/1 (0V or VDD)

• Very rubust! • Inverter (and CMOS) has a large noise margin. • Modern CPU – billions of transistors (logic and memory) • We can keep a logical state for years (if attached to power) • We can easily copy a bit

However – some problems are hard to solve on a classical computer Erik Lind EIT, Lund University 4

Computing Complexity Searching an unsorted list Grovers algorithm. BQP • P – easy to solve and check on a 푛 classical computer ~O(N)

• NP – easy to check – hard to solve on a classical computer ~O(2N) P P • BQP – easy to solve on a quantum computer. ~O(N) NP • BQP is larger then P

Factoring large • Some problems in P can be more numbers – efficiently solved by a quantum Shor’s Algorithm computer Simulating quantum systems Note – it is still not known if 푃 = 푁푃. It is very hard to build a quantum computer… Erik Lind EIT, Lund University 5

Application of QC:s

Modeling of ‘quantum systems’

• Penicillin – 41 atoms in • 1086 bits in a classical computer

• 290 quantum bits

Room temperature (?) Erik Lind EIT, Lund University 6

Application of QC:s Erik Lind EIT, Lund University 7

Your background?

푑 ℏ2 d2 푖ℏ Ψ 푥, 푡 = − + 푉 푥 Ψ 푥, 푡 Schrödinger equation in real space 푑푡 2푚 dx2 representation

푑 푝Ƹ2 푖ℏ |Ψ 푥, 푡 ۧ = + 푉 푥ො |Ψ 푥, 푡 ۧ Schrödinger equation in abstract vector 푑푡 2푚 space (Dirac) notation Erik Lind EIT, Lund University 8

Super-basics of linear algebra

A vector V in 3D space 푉 = 훼푒푥 + 훽푒푦 + 훾푒푧 푒푦

V 푒푥, 푒푦, 푒푧 are the basis vectors (Normalized, orthogonal to each other)

We can represent any vector with three numbers 푒푥

푒 푧 We can chose a different set of basis vectors ′ ′ ′ 푒푥, 푒푦, 푒푧 are the basis vectors

The same vector can then be written with three different numbers 훼′, 훽′, 훾′ ′ ′ ′ ′ ′ ′ 푉 = 훼 푒푥 + 훽 푒푦 + 훾 푒푧 Erik Lind EIT, Lund University 9

Basics of Quantum Mechanics

푥 • Every physical system is a state. 3

푥4 • A state of a system is a vector (ray) in a abstract complex vector (Hilbert) Space. 휙

• Denoted 휙 using Dirac notation. (You can sort of think of this as a vector in nD space.)

• – something we can measure about the particle. • Energy 푥 푥 1 푥 2 • Position 푛 • Momentum • Spin direction 휙 = 훼 푥1 + 훽 푥2 + 훾 푥3 + ⋯ 푉 = 훼푒푥 + 훽푒푦 + 훾푒푧 • Orthonormal basis vectors – these are the • Each set of observables is represented by a set of basis vectors • These are ‘rotated’ with respect to each other

• A state can be in a superposition of the basis vectors. The coefficients are complex numbers 휙 = 훼 푥 + 훽 푥 + 훾 푥 Coefficients 훼, 훽, 훾 are complex numbers 1 2 3 State is normalized: 훼 2 + 훽 2+ 훾 2 + ⋯ = 1 Erik Lind EIT, Lund University 10

Basics of Quantum Mechanics

Examples

• The position of a in a molecule 휙 = 훼 푥1 + 훽 푥2 + 훾 푥3 + ⋯

• Energy of a particle in a quantum well 휙 = 훼 퐸1 + 훽 퐸2 + 훾 퐸3 + ⋯

• Energy of an electron in an atom Example – the position of 0 1 an electron on a molecule • Energy of an electron a harmonic oscillator 0

• The spin on an electron or a neutron. 2 1 5 3 2 4 4 5 3

This would be a 6 dimensional space! Erik Lind EIT, Lund University 11

What about the wavefunction 횿(퐱)?

Ψ 푥

x

ℏ2 d2 • We are used of solving the S.E equation − + 푉 푥 Ψ 푥 = 퐸 Ψ 푥 2푚 dx2 푛 • Energies and wavefunction

Ψۧ = ෍ 푐푛|xnۧ| 푛

• Expansion coefficients cn is the ! • Probability of finding electron at x=xn! • Position space – infinite (and uncountable) dimensions Erik Lind EIT, Lund University 12

Uncertainty Relations

Depending on the choice of basis – the state of an particle can be described using different basis vectors! 푏푛 푏1 푎푛 Example – the position of a electron is known to be x1

푎4 휓 = 푥1 The electron is then in a superposition 휓 = 푎 푝1 + 푏 푝2 + 푐 푝3 + ⋯ 푏4 of different momentums!

푎1 Ex: Difference between position and momentum eigenstates 푏2 푎3 푏3 Uncertainty relation Δ푥Δ푝 ≥ 푖ℏ 푎2 Erik Lind EIT, Lund University 13

Measurements in Quantum Mechanics

Measurement – apparatus which determines an 휓 = 푎1 푥1 + 푎2 푥2 + 푎3 푥3 + ⋯ observable (Energy, position, Spin direction…)

푥푛

푥4

푥1

푥3 A measurement can be though of running a state 푥2 through a machine (apparatus) • A pointer shows the measured value of the During a measurement of an observable a observable State collapses to an eigenstate with probability • The state after the measurement is an eigenstate 푃 = 푎 2 푛 푛 • The outcome is non-deterministic • Not deterministic and NOT time reversible! • Collapse of wavefunction Erik Lind EIT, Lund University 14

Quantum Bits ۧ↓ For quantum computation – effective 2D dimensional Hilbert space This is called a quantum bit (, qbit) ۧ↑| Example - Spin ½ particle - |↑ۧ ↓ۧ : 2 dimensional Hilbert space •

휓푠 = 훼 0 + 훽 1 = 훼 ↑ + 훽 ↓ ห푥2ൿ ห퐸2ൿ Example : Two energy levels of an atom

휓퐸 = 훼 0 + 훽 1 = 훼 퐸1 + 훽 퐸2 ห푥1ൿ ห퐸1ൿ Example : Position of electron in two quantum dots

휓퐸 = 훼 0 + 훽 1 = 훼 푥1 + 훽 푥2

훼 휓 = 훼 0 + 훽 1 훽 a anb b – two complex numbers State is normalized: 훼2 + 훽2 = 1 Vector representation of a state Erik Lind EIT, Lund University 15

Quantum Bit – Bloch sphere

1 0 Normalized Complex coeff. 휓 = 훼 0 + 훽 1 휃 sin 1 푖 휃 휃 훼 2 푌 = 0 + 1 휓 = sin 0 + cos 푒푖휙 1 = 2 2 훽 휃 2 2 e푖휙cos 2 1 1 2 푖

0 1

1 1 푋 = 0 + 1 2 2 1 1 For one spin ½ particle – the Bloch sphere 2 1 sort-of represents the spatial orientation of the spin! Erik Lind EIT, Lund University 16

Quantum System Dynamics

A quantum system evolves in time set by the 휓(푡) = 훼(푡) 푎1 + 훽(푡) 푎2 + 훾(푡) 푎3 + ⋯ Schrödinger Equation, where H is the Hamilton . 푎푛 푑 푖ℏ 휓(푡) = 퐻෡(푡) 휓(푡) 푎4 푑푡 푝Ƹ 2 퐻෡ = + 푉(푥ො) Electron in potential 2푚 푎1 퐻෡ = −훾퐵푧휎푧 Spin in magnetic field 푎3 푎 • This give a set of coupled differential equations for 2 the coefficients a b g…

• By applying an Hamiltonian to a – move the state in the Hilbert space over time. We can in this way control a quantum state.

• This is deterministic and time reversible! Erik Lind EIT, Lund University 17

Quantum Mechanical bit

Operator - 푛 × 푛 matrix

Time evolution U of a state – unitary matrix Preserves norm of quantum state Visualized as rotation of point on Bloch Sphere

푖 퐻푡 휓(푡) = 푒ℏ 휓(0) If the Hamiltonian H is independent of time

By chosing H and t – we can induce any rotation on the Bloch Sphere

훼1 푎 푏 훼0 = All 1-bit gates can be represented by a unitary2x2 matrix 훽1 푐 푑 훽0 Erik Lind EIT, Lund University 18

Single quantum Gate – Bloch Sphere rotation

t Quantum gates are 휓 휓′ reversible! U

Let H(t) act on 휓 for some time t0.

These can in general be build by applying time varying B/E field to a spin/position qbit in x,y and z 휃 휃 direction for a fixed time. 휓 = sin 0 + cos 푒푖휙 1 2 2 NOT gate p rotation around x axis 0 → 1 훼 휓 = 훼 0 + 훽 1 훽 1 Hadamard gate: 0 → 0 + 1 2 Matrix representation of a state

These also work on superpositions: 휓 = 훼 0 + 훽 1 Erik Lind EIT, Lund University 19

Many qubits

A Qubit A Tensor product 퐻퐴 휓 = 0 퐴 ⊗ 1 퐵 = 01 퐻퐵 B Qubit B Example: Two non-interacting qubits – product state

휓 퐴 = 훼퐴 0 퐴 + 훽퐴 1 퐴 휓 = 휓 퐴 ⊗ 휓 퐵 = 훼퐴훼퐵 00 + 훽퐴훼퐵 10 + 훼퐴훽퐵 01 + 훽퐴훽퐵 11

휓 퐵 = 훼퐵 0 퐵 + 훽퐵 1 퐵 Example: Two correlated qubits – entangled state 1 휓 = ( 10 + 01 ) This can NOT be written 2 as a product state!

ۧ N Coupled qbits requires 2푁 complex Example: General 2 qubit state ↓↓ numbers to describe 휓 = 훼 00 + 훽 10 +훾 01 +훿 11 ۧ↑↑| ۧ↓↑ N=100 : 2100 ≈ 1030 complex numbers

ۧ↑↓| Erik Lind EIT, Lund University 20

Equal superposition of qubit states

1 휓 = 0 + 1 1 qubit: 2 states 2 1 휓 = 00 + 10 + 01 + 11 2 qubit: 4 states 4

1 휓 = 000 + 001 + 010 + 100 + 011 + 110 + 101 + 111 8 states 8

휓 = ⋯ . 100 spin ½ : 2100 = 1030 dimensional space

• There are 1080 atoms in the observable 휓 = ⋯ . 300 spin ½ : 2300 = 1090 dimensional space universe

• Hilbert spaces are big!

A quantum state can represent a huge number of 0’s and 1’ simultaneously! Erik Lind EIT, Lund University 21

Measurement of multiple qubit states

푁 = 100 ↔ 1030 푠푡푎푡푒푠 1 휓 = 000 … + 001 … + 010 … + 100 … + 011 … + 110 … + ⋯ 2푁 Will randomly select ONE of the states with equal probability!

Gives randomly 100 bits of information or out 1030 Direct application of quantum parallelism is not an easy task!

1 휓 = 000 … + 001 … + 010 … + 휓 = 010 … 2푁 0

Picks ONE of states of |휓ۧ Measure A Erik Lind EIT, Lund University 22 Entanglement – ”spooky action” at a distance

1 휓 = ( 10 + 01 ) 2 A B A B Qubit A Qubit B

1) Create entangled qbit state 2) Separate qubits 3) Measurement on qbit A 1 directly sets state of qbit B. Before measurement: 휓 = ( 10 + 01 ) This has been 2 experimentally verified. 4) Directly = much faster then speed of light at least.

Reality might be non-local. Erik Lind EIT, Lund University 23 Entanglement – ”spooky action” at a distance

1 휓 = ( 10 + 01 ) 2 A

Qubit A Qubit B BB

1) Create entangled qbit state 2) Separate qubits 3) Measurement on qbit A 1 directly sets state of qbit B. Before measurement: 휓 = ( 10 + 01 ) This has been 2 experimentally verified. After measurement 휓 = 10 (50% chance) 4) Directly = much faster then on particle A only: speed of light at least. We now know that a measurement on B will yield Reality might be non-local. 0 with 100% accuracy! Erik Lind EIT, Lund University 24

Quantum Gates – Controlled Gate

Let H(t) act on y for some time t0 only if x=1

휓 휓′ 푦 푥 U 푥 푦 푈 푦 or 푦

If x=0 do nothing to qubit y 푦 If x=1 apply rotation U to qubit y

Example: CNOT Gate 휃 휃 휓 = sin 0 + cos 푒푖휙 1 2 2 If x = 0 then 푦 = 푦 If x = 1 then apply NOT to 푦

• 2 qubit gate – can be represented by a 4x4 matrix

• Controlled Gates also work on super positions – quantum parallelism! Erik Lind EIT, Lund University 25

Universial set of quantum gates

These set of gates are ‘universal’ Can implement all logic functions NAND, NOR etc.

1 qbit gate – controlled rotation of Bloch Sphere

Controlled Gates – almost anyone will do, controlled NOT (CNOT) or controlled phase CPHASE most common.

휓 = 훼 00 + 훽 10 +훾 01 +훿 11 훼 훽 훾 훿 Erik Lind EIT, Lund University 26

Example - Deutsch’s Algorithm

푓 푥 =? x fc(x) fb(x) fb(x) fb(x) 푥 푓 0 ≠ 푓 1 “balanced” Uf 푥 0 0 0 1 1 푦 푦⨁푓(푥) 푓 0 = 푓 1 “constant” 1 0 1 0 1 c b b c 푓(푥) takes a single bit {0,1} into a single output bit {0,1}.

If we don’t know if f(x) is “balanced” or “constant” – but would like to know

Classical Computer – calculate f(0) and f(1) and compare – need to evaluate f twice!

The below can achieve this by evaluating f(x) only once!

0 퐴 H Uf H Measure qubit A 0 푓 0 = 푓 1 1 푓 0 ≠ 푓 1 1 퐵 H Erik Lind EIT, Lund University 27

Example –

The quantum circuit below allows for an unknown qubit A in a state !휓ۧ = 훼|0ۧ + 훽|1ۧ to be recreated (‘teleported’) unto qubit C|

Entangels A, B and C Measurement 0 or 1 A 휓 H M 0 or 1 B 0 H M Apply X if 1 C 0 Z X 휓 Apply Z if 1 Entangels B and C

Note that my measuring A and B – we only obtain 0 or 1! Erik Lind EIT, Lund University 28

General

x0 x0 Input x1 x1

x … Gates which calculates 2 x2 … … 0 f0 푥 0 → 푥 푓(푥) 0 f1 0 Output f2 f(x) … … 2푁−1 1 Equal superposition of all N inputs: This can be done with N Input register ෍ 푥 2푁/2 Hadamard gates and CNOT gates 0 By just computing the f(x) once, f(x) is 2푁−1 2푁−1 f(x) 1 “calculated” for all 2N values! ෍ 푥 0 → ෍ 푥 푓(푥) 30 2푁/2 N=100 – 10 speedup! 0 0

2푁−1 1 However, a direct measurement of x 푁/2 ෍ 푥 푓(푥) → 푥0 푓(푥0) 2 gives only ONE random value of f(x0)! 0

Measurement of 푥 Need clever algorithms similar to Deutsch Problem to try and extract more information from the correlation between x and f(x) Erik Lind EIT, Lund University 29

Decoherence – random Bloch sphere rotation

A single qbit will couple to the environment (example for a single ½ spin)

• Nuclear spins • Stray B-fields

• Electrical fluctuations Couples through spin- • Interactions with photons orbit interactions • Electrical Noise

This will induce a random 퐻(푡) which shifts around the quantum state.

Quantum states can be very fragile. pS – mS before the qbit is completely random. Decoherance time (T1 / T2). Erik Lind EIT, Lund University 30

Decoherence – random Bloch sphere rotation

• T1 – Spin relaxation Time E1 • T2 - Decohrerance Time T1 Given two distinct energy states for E0

If particle initially in E1 – how long time until it is found in E0? y T1 – Spin relaxation time

If particle in a superposition - how long until the phase relation between is lost? x T2 – decoherence time

These are measures of the ‘quality’ of a qbit. We need to do our gate operations much shorter then T1 and T2. Erik Lind EIT, Lund University 31

Quantum Computers - implementations

– Spin qubits (compatible with Si processing!) – Optical traps (Most stable!) – Superconducting Qubits (Most qubits!) – Majorna Zero Modes – Topological Qubits (Most difficult to understand.) Erik Lind EIT, Lund University 32

Ion Trap – Optical Qubits

Atomic Ions trapped in RF/DC field Laser cooling Strongly isolated from environment +Very long decoherance T1 times (s- years(!!) -Scalability? -Geometry?

10µm Qubit Operation B-field + 729 nm laser – excite between 0 and 1

Multi-qubit operation motivated through phonons

Readout – 397nm (and 866nm) laser Scattered light only if atom in state 0 Erik Lind EIT, Lund University 33

Spin Qubits

Single ½ spin in B-field 1 E2

Two energy eigenstates – Spin up and spin down ∗ Δ퐸푧 = 푔 휇푏퐵 Energy split is small

* B=1T (InAs, g =-10) 0 E1 gives Δ퐸푧 = 0.5 푚푒푉

Thermal energy (~kT/q) should be smaller then this value

T=300K - 25 meV T=1K – 0.08 meV

We need to operate our qubits at cryogenic temperature! Erik Lind EIT, Lund University 34

Spin Qubits Manipulation

BZ 1 E A static magnetic field in 2 the Bz direction – two ∗ level system Δ퐸푧 = 푔 휇푏퐵

RF magnetic field in x direction – can manipulate spin superposition 퐵푥(푡) 0 E1

Resonance – only effective is RF frequencies close to energy difference Can access different qubits by are effective in rotating the global B-field. Single qubit spin! control!

∗ ℏ휔푅퐹 = Δ퐸푧 = 푔 휇푏퐵 This is very similar to Nuclear magnetic resonance Erik Lind EIT, Lund University 35

Spin Qubit Implementation

Single Electron Transistor – can store single

Bloch sphere rotation in a Si SET

T1 ~ 150 nS Erik Lind EIT, Lund University 36

Decoherence

Single Electron will interact with a background of many nuclear spins

These are randomly distributed in the material nS decoherence time

Electron moving in a electric field will experience Spin qubits a magnetic field. This leads to spin-orbit- Materials with no nuclear spin interaction. Electrical noise (and phonons!) can • Si29 thus couple to a spin. • C12 Spin-orbit-interaction is stronger for heavy elements (III-Vs) Light elements (C, Si – low SOI) Erik Lind EIT, Lund University 37

Spin Qubit Si29, Si29-P

Electron quantum dot Spin qubit T2=268 µS

Electron – Donor Qubit T2=567 µS

Nuclear Spin Qubit T2= 0.6S

Electron Spin This can be increased to 푇2 = 30푆 (!!) using pulse focusing techniques! T1 >> 100 µS (!!) Erik Lind EIT, Lund University 38

Two Spin Qubit Gates

Two coupled qubits

By tuning the coupling – implementation of CNOT gate!

퐻 = −퐽푆1 ∙ 푆2

Two nearby spins will influence each other through exchange mechanisms!

Charge sensor (Spin detector) Erik Lind EIT, Lund University 39

Spin Qubit Implementation

Spin qubit summary

+Can be made with standard Silicon Technology (!!)

+Small – 10 – 100 nm size

+Long decoherance times can be demonstrated

+CNOT gate

+ GHz resonance frequencies - fast

- Sensitive to single atomic defects – difficult to control - How to couple many qubits?

Currently - at a single qubit / single gate stage. Erik Lind EIT, Lund University 40

Superconducting Qbits Erik Lind EIT, Lund University 41

Superconductors

• Some metals becoms superconducting at low T Al – TC=1.12K Again – we need to operate Nb – TC= 9K qubits at mK temperatures! • Really Zero resistivity! – Induced current decay over very long times (millions of years..)

Δ • BCS Theory EF – Electrons bound to form Cooper pairs (↑+↓) – Energy-gap around the Fermi energy – Can move through the metal without any resistance

• Macroscopic wavefunction

– ns – Cooper pair concentration A macroscopic superconductor behaves 휓(푥) = 푛 (푥)푒푗 휙 푥 – 휙– global phase 푠 electrically as a single quantum system. Erik Lind EIT, Lund University 42

Quantum LC-circuit

Stored energy in C Stored energy in L 휙 = න 푑푉 1 1 1 푄, 휙 = −푖ℏ 휔 = 퐻 = 푄2 + 휙2 퐿퐶 2퐶 2퐿 1 1 Energy oscillates 퐻 = 푝2 + 푚∗휔2푥2 푝, 푥 = −푖ℏ between capacitor 2푚∗ 2 and inductor The quantum version of the LC-circuit behaves as an Harmonic Oscillator!

… Equi-distant energy separation between 3 energy levels 2 Δ퐸 = ℏ휔 1 0 Not a two-level system! Erik Lind EIT, Lund University 43

Josephson Junction – nonlinear inductor

푗 휙2 휓(푥2) = 푛푠(푥2)푒 Superconductor Insulator 푖푗 푖푗(푡) = 퐼0sin(휙2 − 휙1) 훿 is the phase difference between ℏ 푑훿(푡) 푑휙 the two superconductors. 푣퐽(푡) = = + 푣푗 − 푒 푑푡 푑푡 Superconductor

푗 휙1 푑푖 푑훿 푑훿 푑푖 1 푒 1 푑푖푗 휓(푥1) = 푛푠(푥1)푒 푗 푗 = 퐼0 cos 훿 = 푣푗 푡 = 푑푡 푑푡 푑푡 푑푡 I0cos 훿 ℏ I0cos 훿 푑푡

푑푖 푣 = 퐿 Ordinary inductor 퐿 푑푡

• A JJ-junction will act as a non-linear inductor • L increases when the stored flux increases • This will lead to anharmonicity LC oscillator with different energies Erik Lind EIT, Lund University 44

SC qubit - transmon

Nonlinear L

Slightly different L for higher energies 휔01 ≠ 휔12 Two level system!

푑휙 = 푣 푑푡 Erik Lind EIT, Lund University 45

Tunable Josephson Junction - SQUID

퐼 depends on how well the wave function overlaps 푖 (푡) = 퐼 sin(훿) 0 푗 0 in the insulating region. Can be difficult to control.

Solution – connect two JJ and apply an magnetic field. B This can be shown to change 퐼표 → 퐼0cos(푘퐵). A local B-field can be created by running a DC-current through a conductor

We get a JJ-junction which can be tuned to adjust the non-linear L!

idc We can thus tune the resonance frequency of the qubit! Erik Lind EIT, Lund University 46

Superconducting Qubit

Nonlinear L This forms an isolated island for charges ‘Coper-pair box’ Sensitive to charge fluctuations & electrical noise

Solution – make C large! Variation in charge on the box – small change in energy C Insensitive to charge noise – longer time!

Transmon Qubit

1 C JJ Junction 0 Transmission Line Shunted Plasma Oscillation Qubit Erik Lind EIT, Lund University 47

Superconducting Qubit Control

푣푎푝푝 푡 = 퐴(푡)푐표푠(휔푑 − 휙) Capacitviely couple a RF signal to the transmon

• Frequency difference - 휔푞 − 휔푑 rotation around z-axis • Phase difference – rotation around x and y axis

• Full control over Qbit states!

• 휔푞 is in the GHz range – nS time / operation Erik Lind EIT, Lund University 48

Superconducting Qubit Control

푣푎푝푝 푡 = 퐴(푡)푐표푠(휔푑 − 휙)

• 휔퐿푂 is in the 1-10 GHz range • I/Q signals are in the nS range. Erik Lind EIT, Lund University 49

Superconducting Qubit Readout

Waveguide

휔 푟푒푎푑 ?

푣푎푝푝 푡 = 퐴(푡)푐표푠(휔푑 − 휙)

To read out a qubit: • Couple it (weakly )to a waveguide resonator • Measure the resonance frequency (away from qubit frequency!) • The state of the qubit will perturb the resonator Erik Lind EIT, Lund University 50

Superconducting Qubit Readout

Qubit L1 varies if 표푟 |1ۧ 0ۧ|

1ۧ|

0ۧ| Erik Lind EIT, Lund University 51

2-Qubit Gates

• Qbit interaction is possible by coupling different qubits together

• Interaction can be controlled by tuning coupling strength or qubit frequency.

• This can be tunable through coupling via a DC SQUID

• CPHASE/CNOT qubit control can be implemented Erik Lind EIT, Lund University 52

Google Quantum Supremacy (?) Erik Lind EIT, Lund University 53

Superconducting Qubit Control

Transmon Ion Trap Current state-of-the-art

Google Bristlecone – 72 qubits

Superconducting qubits – reasonable advanced circuits 50-75% success rate Erik Lind EIT, Lund University 54

Superconducting Qubit Control

Sources of Decoherance

• Emission of photon

• Critical current flucutations (I0 in JJ) • Dielectric losses • Quasi-particle tunneling

• Charge noise

Decoherence Times in the 10-100 µS time range. Erik Lind EIT, Lund University 55

Topological Qbits

• All qbits are sensitive to decoherence (nS-mS decoherence time) • Local flucutations causes decoherance

• Can we build a qubit which is insensitive towards local pertubations? – (Microsoft thinks so!)

• These are so called Majorana Fermions – ‘Topological protected states’

• Should appear in a “p-type” superconductor • Cooper pairs with same spin • Can form at the edge of a p-type superconducting wire

• p-type superconductors do not really exist…. Erik Lind EIT, Lund University 56

Majorana Zero Modes

B-field Superconductor with epitaxial contact to nanowire

Nanowire (1D)with large spin-orbit interaction and large g-factor (InSb, InAs). • Under very precise conditions we can then form Majorana Zeros Modes (Majorana Fermions) at the ends to the nanowire. (Very low T, exact position of Fermi energy, high quality SC-N interface, thin SC…)

• These are part of the same (quasi-)particle • The 1D nanowire will be spin polarized • Highly non-local! • Superconductivity can be induced from • To perturb the particle state one needs to perturb both the standard superconductor ends of the wire – unlikely!

• The nanowire system approximates a • ‘Topological protected states’ p-type superconductor! • Potentially very long coherence times! Erik Lind EIT, Lund University 57

Majorana Zero Modes

Ordinary S-N S-N tunneling tunneling junction junction with a MZM

Δ/2 Superconducting Δ The Majorana state E Δ gap EF is formed in the gap F

Need to apply D/2 in Vds for a current to flow!

Zero Bias Peak (Strong) indication that MZM have been formed

This is what has been experimentally shown to far… Erik Lind EIT, Lund University 58

MZM qubit gates (?)

• The MZM adhers show ‘non-abelian’ statistics

• Interchange of two particles change the particle state in a non-trivial way • i.e. not only a change of phase

• Apply network of wires and gates to move Majorana states around each other.

• Could implement some qubit gates.

• This has never been demonstrated. It is not clear which is the best way to implement • (Noble prize?) Qubit or Gates… Erik Lind EIT, Lund University 59

MZM qubit gates

• Measurement could be done by ‘fusing’ two states • Its own anti-particle – will annihilate

• Nothing or creation of electron

• This has also never been demonstrated. Erik Lind EIT, Lund University 60

Quantum Computing Conclusions

• Quantum Computers CAN do SOME things much better then a classical computer will be able to.

• Some IMPORTANT problems can get an EXPONENTIAL speed up

• Qubits are DIFFICULT to build

• Quantum Gates are even HARDER to build

• Plenty of very interesting and deep physics!

• Plenty of very interesting engineering challanges!

• There are alot of large companies (Google, Microsoft, Intel, IBM) working on this! Erik Lind EIT, Lund University 61

Quantum Computing Meme Name/ context 62