Demonstration of two-qubit algorithms with a superconducting quantum processor

Elisa Wall and Sajanth SubramaniamETH Zürich

Spring Semester 2018

Introduction

Requirements and tasks of a Quantum Processor:

Introduction

Requirements and tasks of a Quantum Processor:• State Preparation

Introduction

Requirements and tasks of a Quantum Processor:• State Preparation

• Long coherence time

Introduction

Requirements and tasks of a Quantum Processor:• State Preparation

• Long coherence time• Universal Gate operations

Introduction

Requirements and tasks of a Quantum Processor:• State Preparation

• Long coherence time• Universal Gate operations• Read out of qubits

First solid-state quantum processor

L.DiCarlo et al.

Overview

Two – qubit interaction via

cavity

Avoided crossing Conditional phase gate

Quantum algorithms

Chip design• Two transmon qubits QL and QR

• Voltages VL and VR used to tune qubittransition frequencies fL and fR via generatedflux Φ𝐿𝐿,𝑅𝑅:

where Φ0 is the flux quantum

• Qubits coupled to a microwave cavity, letting qubits couple via virtual photon exchange

• Read-out via homodyne detection

L.Dicarlo et al.

Avoided Crossing in the Cavity

For single atom in cavity we have according to Cavity QED:

Jaynes – Cummings Hamiltonian

For we get the (unnormalized) Eigenstates:

wikimedia

Dressed states

Avoided Crossing in the Cavity

Jaynes – Cummings Hamiltonian

with

In general:

Rabi splitting

Qubit Frequency

Ener

gy

For single atom in cavity we have according to Cavity QED:

For we get :

Avoided Crossing between Qubits

For two transmon qubits in a cavity we have in the dispersive limit

an additional effective interaction term (in second order pertubation) 1,2:

1. A. Blais et al.,»Quantum-information processing with circuit quantum electrodynamics » Phys. Rev. A 75, 032329 (2007)2. J.Majer et al. , » Coupling Superconducting Qubits via a Cavity Bus », arXiv:0709.2135v1 (2007)

Coupling between the cubits via virtual photons in the cavity1

Avoided crossing between frequencies near

Single Qubit Spectroscopy

L.DiCarlo et al.

Point I:sweet spot of strong dispersiveness:a) state preparationb) single qubit rotations andc) measurements

Point II:working point to get to two-bit gates as conditional phase gate

Point IV:𝑄𝑄𝑅𝑅 tuned in resonance with cavity get qubit-cavity interaction strength g

Point III:𝑄𝑄𝑅𝑅 tuned in resonance with 𝑄𝑄𝐿𝐿 get qubit-qubit transverse interaction strength J

Single and Two-Excitation Spectrum

L.DiCarlo et al.

L.DiCarlo et al.

Dynamic Phase

Qubit wavefunction accumulates phase when undergoing frequency change over time

with being deviation from the qubit frequency at point I

Realization of a C-Phase Gate• Cavity mediated interaction results in lowering of

frequency w.r.t. sum of and :

• This phase shift is due to the avoided crossing• We can use adiabatic flux pulses produce phase gates via

by varying with deviationfrom its frequency value at starting point I

• For 𝜙𝜙01: varying the rise and fall of the 𝑉𝑉𝑅𝑅 pulse• For 𝜙𝜙10: varying the amplitude of a simultaneous

weak 𝑉𝑉𝐿𝐿 pulse• For 𝜙𝜙11 :

L.DiCarlo et al.

• Aim: generating phase gates by creating all even and only one odd multiple of 𝜋𝜋 in the phases

• Example: Can implement a conditional phase gate (C-Phase gate 𝑐𝑐𝑐𝑐11) by adjusting 𝜙𝜙01 and 𝜙𝜙10 to zero and

L.DiCarlo et al.

Realization of a C-Phase Gate

Creating Bell states

.

Single qubit gates• via microwave pulses resonant with 𝜔𝜔𝐿𝐿,𝑅𝑅

Conditional phase gates cU:

Example: Creating the first Bell state

Creating Bell states

.

Single qubit gates• via microwave pulses resonant with 𝜔𝜔𝐿𝐿,𝑅𝑅

Conditional phase gates cU:

Analogously create other three Bell states with 𝑐𝑐𝑐𝑐𝑖𝑖𝑖𝑖:

Read-Out process: Joint Dispersive Readout

• State tomography:• Constructing the 16 entries of maximum likelihood density matrix 𝜌𝜌𝑚𝑚𝑚𝑚

• Turning into measurement basis

• Pulsed measurement of VH yields following measurement operator1

where the |𝛽𝛽𝑖𝑖| are of comparable magnitude when operating in dispersive regime

• Complete set of 15 linear independent operators

• Generating ensemble average by executing sequence 450’000 times

1. Filipp, S. et al. “Two-qubit state tomography using a joint dispersive read-out”. arXiv:cond-mat/0812.2485(2008(

Read-Out: Evaluation of 𝜌𝜌

• Concurrence C• (≡ degree of entanglement) • Maximally entangled for C = 1.0, highest eigenvalue 𝜆𝜆1 of

density matrix

• Purity P• (≡ degree of mixture)

• Fidelity F• (≡ degree of agreement with expected result )

Read-Out of Bell states

• Averaging real part of maximum-likelihood of density matrix 𝜌𝜌𝑚𝑚𝑚𝑚

• Concurrence ≈ 0.81 - 0.94

• Purity ≈ 0.79 - 0.92

• Fidelity to ideal Bell state ≈ 0.87 - 0.94

Algo: Grover’s search

• Aim: finding 𝑥𝑥0 element of computational basis• Classically: 𝑂𝑂(𝑁𝑁), QM: O( 𝑁𝑁)

• Oracle:

• b) – e) Creating Bell state• f) Undoing entanglement with 𝑐𝑐𝑐𝑐00• g) Output state

One iteration over Grovers algorithm:

Algos: Deutsch- Josza Algo

• Aim: find out whether unknown function isconstant [𝑓𝑓0 𝑥𝑥 = 0 (𝑏𝑏) 𝑜𝑜𝑜𝑜 𝑓𝑓1 = 1 (c)]or balanced [𝑓𝑓2 𝑥𝑥 = 𝑥𝑥 (𝑑𝑑) 𝑜𝑜𝑜𝑜 𝑓𝑓3 = 1 − 𝑥𝑥 (e) ]

• Classically: 𝑁𝑁2

+ 1 calls of 𝑓𝑓(𝑥𝑥), QM: one call of 𝑓𝑓(𝑥𝑥)

• Oracle:

• Encoding result in final state of 𝑄𝑄𝐿𝐿 , while leaving 𝑄𝑄𝑅𝑅 untouched• Only and no should be output, and corresponds

to constant and to balanced input function 𝑓𝑓(𝑥𝑥)

Fidelities of Algorithms

Fidelities of Grovers output ≈ 0.80 - 0.82Fidelities of Deutsch-Josza output ≈ 0.84 – 0.93

Performance and Outlook

DiVincenzo's criteria:

• The ability to initialize the state of the qubits to a simple fiducial state.

• Long relevant decoherence times.

• A “universal” set of quantum gates.

• A qubit-specific measurement capability.

• A scalable physical system with well characterized qubits

David P. DiVincenzo, «The Physical Implementation of Quantum Computation», arXiv:quant-ph/0002077 (2000)

Conditional phase gate ( together with one qubit gates) is universal

~ 1 micro second coherence time; allowing for ~10 sucessive gates

Presented two-qubit device:

Dispersive read-out

Via timed pulses with frequncies fL or fR

Problems with scalabitily

Performance and Outlook

Tunable qubits Fixed qubits

• Fast multi-qubit interactions

• Minimal residual coupling

• Flux noise decoherence

• Computational basis leakage

• High coherence time

• Frequency crowding

Negative effects worsen when scaling up the system!

Possible solution: Periodic modulation of qubit frequencies

Performance and Outlook

Performance and Outlook

Tunable two-qubit processor (entangler):

Parametric approach: IBM QX4*(5 Qubits):

Two-qubit gate fidelity

*https://quantumcomputingreport.com/scorecards/qubit-quality/