Quantum State Protection and Transfer using Superconducting Qubits Dissertation Defense of Kyle Michael Keane Department of Physics & Astronomy Committee: Alexander Korotkov Leonid Pryadko Vivek Aji June 29, 2012

Journal Articles 1.A. N. Korotkov and K. Keane, “Decoherence suppression by quantum measurement reversal,” Phys. Rev. A, 81, (R), April K. Keane and A. N. Korotkov, “Simple quantum error detection and correction for superconducting qubits,” arxiv: , May 2012 (submitted to Phys. Rev. A). 1.“Decoherence suppression of a solid by uncollapsing,” Portland, OR, March 15-19, 2010), Z “Currently realizable quantum error correction/detection algorithms for superconducting qubits” Dallas, TX, March 21-25, 2011), Z “Modeling of a flying microwave qubit” Boston, MA, Feb. 27-March 2, 2012, Y APS March Meeting Presentations Posters

Outline 1.Introduction 2.Decoherence by uncollapsing Korotkov and Keane, PRA Repetitive N-qubit codes and energy relaxation Keane and Korotkov, arxiv: , submitted to PRA, Two-qubit quantum error “correction” and detection Keane and Korotkov, arxiv: , submitted to PRA, Qubit state transfer Keane and Korotkov, APS March Meeting, Summary

INTRODUCTION Let’s begin with a basic

 U δ δ quantum variable Superconducting Phase Qubits state control I μw flux bias I b meas. pulse I meas SQUID readout I sq V sq 25 mK SQUID flux bias qubit C I0I0 L  microwaves X-, Y-rotations flux bias Z-rotations operation  U ΔUΔU |0|0 |1|1

State Measurement SQUID-based Measurement: lower barrier for time t  U relaxes |1|1 |0|0 readout w/ SQUID Tunneling Detected = state has been projected onto |1  and destroyed Tunneling Not Detected = state has been projected onto |0 

Weak Measurement lower barrier for short time t  U relax |1|1 |0|0 readout w/ SQUID Tunneling Not Detected = state projected onto|0  OR state was |1  and didn’t have enough time to tunnel There is a small change to the energy spacing during the lowering of the barrier Tunneling Detected = state has been projected onto |1  and destroyed

Uncollapsing If tunneling does not occur, the qubit state is recovered In experiment, only data for cases where tunneling does not occur is kept State Prepared Doesn’t Tunnel Partial Measurement Projects state toward 0 (was 1) Partial Measurement Projects state toward 0 π-pulse

Zero-Temperature Energy Relaxation This can be “unravelled” into discrete outcomes with probabilities |0|0 |1|1 The population of the excited state moves into the ground state

DECOHERENCE SUPPRESSION BY UNCOLLAPSING Project One Korotkov and Keane, PRA 2010

Protection from Energy Relaxation Quantum Error Correction (Shor/Steane/Calderbank circa 1995) Requires larger Hilbert space and controllable entanglement) Decoherence-Free Subspaces (Lidar 1998) Requires larger Hilbert space and specfic subspaces Dynamical Decoupling (Lloyd and Viola 1998) Does Not Protect Against Markovian Processes (Pryadko 2008) Standard methods to protect against decoherence: Our proposed method Simple modification of uncollapsing procedure Our proposal was demonstrated in another system Requires selection of only certain cases Similar to probabilistic QEC and linear optics QC

Ideal Procedure  11 Preparedπ-rotation Partial meas. (p u ) Partial meas. (p) π-rotation time axis of π-rotation Initial value Returned to initial value Similar protection for all density matrix elements Korotkov and Keane, PRA 2010

Results Yields a state arbitrarily close to initial Some improvement even with naive uncollapsing strength Korotkov and Keane, PRA 2010 Fidelity Measurement Strength (p)

Process with Decoherence storage period t  11 Preparedπ-rotation Partial meas. (p u ) Partial meas. (p) π-rotation time axis of π-rotation Initial value Pure dephasing and energy relaxation during entire process Returned to initial value Korotkov and Keane, PRA 2010

Results Pure dephasing uniformly decreases fidelity Explains phase qubit uncollapsing experiment (Katz, 2008) Still works with relaxation during operations Perfect suppression requires small prob. of success Korotkov and Keane, PRA 2010 Fidelity and Probability Measurement Strength (p)

Experimental Demonstration Weak Measurement polarization beam splitter, half wave plate, and dark port Optical CircuitResults Nearly exact match to theory Jong-Chan Lee, et. al., Opt. Express 19, (2011) Relaxation similar components, (except no dark port)

Protecting Entanglement Initially entangled state Q1 Q2 WMπ π Entanglement is recovered Q1 Q2 WMπ π Circumvents Entanglement Sudden Death Same optics group did this extension experiment Yong-Su Kim, et. al., Nature Physics, 8, (2012)

Summary Does not require a larger Hilbert space Modification of existing experiments in superconducting phase qubits Demonstrated using photonic polarization qubit Extended to protect entanglement

REPETITIVE CODING AND ENERGY RELAXATION Project Two Keane and Korotov, arxiv 2012

Motivation Bit Flip A bit flip looks like a more difficult error process than T1 T1 AND Repetitive coding protects against bit flips PROTECTS ????????? THEREFORE…

Repetitive Quantum Codes and Energy Relaxation |  | 0 N-1  tomography T 1 (i) X X All “N-1” are 0: good Any in 1: either discard (detection) or try to correct (correction) Encoding by N c-X gates |  | 0 N-1  X |  |0|0 |0|0 |0|0 |0|0 c-X gate (cNOT) Syndrome Result FAILS

Two-Qubit Encoding Two qubits Equal decoherence strength T 1 (i) Keane and Korotov, arxiv 2012 Fidelity Decoherence Strength (p)

N-Qubit Error Detection |  | 0 N-1  tomography T 1 (i) X X All “N-1” are 0: keep Any in 1: discard p Keane and Korotov, arxiv 2012 Fidelity Decoherence Strength (p) ignore detect single

N-Qubit Error Correction |  | 0 N-1  tomography T 1 (i) X X All “N-1” are 0: keep Any in 1: cannot correct! p QEC is impossible In our paper we show that no unitary operation can improve the fidelity for p<0.5 Keane and Korotov, arxiv 2012 Fidelity Decoherence Strength (p) ignore correct single

Summary Can be used for QED, but not for QEC of energy relaxation 3 qubits are optimal, but 2 qubits are sufficient

TWO-QUBIT QUANTUM ERROR DETECTION/CORRECTION Project Three Keane and Korotov, arxiv 2012

Two-Qubit Error “Correction”/Detection 0: good 1: either discard (only detection) or correct (if know which error) Y/2 -Y/2 |  |0|0 tomography X-correction needed Y-correction needed Z-correction needed no correction needed (insensitive) Notations: = c-Z E1E1 E2E2 E 1 = X-rotation of main qubit by arbitrary angle 2  : E 1 = Y-rotation of main qubit: E 2 = Z-rotation of ancilla qubit: E 2 = Y-rotation of ancilla qubit: good Keane and Korotov, arxiv 2012

Two-Qubit Error “Correction”/Detection 0: good 1: either discard (only detection) or correct (if know which error) Y/2 -Y/2 |  |0|0 tomography Notations: = c-Z E1E1 E2E2 Various Decoherence Strengths Fidelity Rotation Strength (2θ/π) corr det ign All Four Errors Fidelity Rotation Strength (2θ/π) corr det ign Keane and Korotov, arxiv 2012

QED for Energy Relaxation store in resonators 0: good 1: discard Y/2-Y/2 |  |0|0 tomography Notations: = c-Z T1T1 T1T1 Y/2-Y/2 QED of real decoherence The fidelity is improved by selection of measurement result 0 Fidelity Relaxation Strength detect ignore Keane and Korotov, arxiv 2012 Almost “repetitive”

Summary QEC is possible for intentional errors QED is possible for energy relaxation Experiments can be done with superconducting phase qubits

QUANTUM STATE TRANSFER Project Four Keane and Korotov, APS 2012

System Resonator or Phase Qubit Transmission Line Tunable CouplersHigh-Q Storage Initially here Sent here Superconducting Waveguide Example from UCSB Tunable Parameter 1 0 Korotkov, PRB 2011

Ideal Procedure Transmission Coefficients Qubit initially is here Qubit transferred to here Time (t) Typical parameters (UCSB) Desired Efficiency Korotkov, PRB 2011

Main idea A B Transmission lineReceiving resonator “into line” “into resonator” AB Korotkov, PRB 2011

Procedural Robustness Transmission Coefficients Time (t) Keane and Korotkov, APS 2012

Shaping of Control Transmission Coefficients Time (t) Robustness No Problem! Keane and Korotkov, APS 2012

Switching Time Transmission Coefficients Time (t) Robustness No Problem! Keane and Korotkov, APS 2012

Maximum Transmission Coefficient Transmission Coefficients Time (t) Robustness No Problem! ( experiments have good control of tunable coupler ) Keane and Korotkov, APS 2012

Frequency Mismatch

Robustness Requires Attention ( resonator frequencies should be kept nearly equal throughout procedure ) Keane and Korotkov, APS 2012

Summary Robust to procedural errors (timing, shaping, maximum transmission coefficient) Requires active maintenance of nearly equal resonator frequencies The second conclusion is very important for experiments — For the current solid-state tunable couplers there is an effective frequency shift during modulation of the transmission coefficient

CLOSING REMARKS recapitulation

Summary Decoherence suppression by uncollapsing – Probabilistically suppresses Markovian energy relaxation – After our proposal, it was demonstrated by another group – Extended in another experiment to entangled qubits N-qubit repetitive codes and relaxation – Can be used for QED, but not for QEC (2 qubits are sufficient) Two-qubit “QEC”/QED experiments – Can be performed with current technology Quantum state transfer – Robust against procedural errors – Requires resonator frequencies to be kept nearly equal THANK YOU!

APPENDICES Just in case

Representations of Errors-Example: Energy Relaxation From the normalization requirement Need to derive this from commutator!!!!! Need to derive this from somewhere!!!!! Solving these equations and combining into an operation Choosing a specific operator sum decomposition If you initially have a pure state, the classical mixture created by this process becomes explicit LINK This can be done for any operation however only some give physically meaningful interpretations Master Equation RETURN