demonstration of two-qubit algorithms with a ... · demonstration of two-qubit algorithms with a...
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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/