Schrdinger Cats Maxwells Demon and Quantum Error Correction

Schrödinger Cats, Maxwell’s Demon and Quantum Error Correction Theory Experiment SMG Liang Jiang Leonid Glazman M. Mirrahimi ** Michel Devoret Luigi Frunzio Rob Schoelkopf Andrei Petrenko Nissim Ofek Reinier Heeres Philip Reinhold Yehan Liu Zaki Leghtas Brian Vlastakis +…. . Quantum. Institute. yale. edu Shruti Puri Yaxing Zhang Victor Albert** Kjungjoo Noh** Richard Brierley Claudia De Grandi Zaki Leghtas Juha Salmilehto Matti Silveri Uri Vool Huaixui Zheng Marios Michael +…. . 1

Quiz (with answers): Q: Is quantum information carried by waves or by particles? A: Yes! Q: Is quantum information analog or digital? A: Yes! 2

Quantum Computing is a New Paradigm • Quantum computing: a completely new way to store and process information. • Superposition: each quantum bit can be BOTH a zero and a one. • Entanglement: Bits can have non-classical correlations. (Einstein: ‘Spooky action at a distance. ’) • Massive parallelism: carry out computations that are impossible on ANY conventional computer. Daily routine engineering and calibration test: Carry out spooky operations that Einstein said were impossible! 3

000, 001, 010, 011, 100, 101, 110, 111 The Power of Quantum Information Even a small quantum computer of 50 qubits will be so powerful its operation is difficult to simulate on a conventional supercomputer. Quantum computers are good for problems that have simple input and simple output but must explore a large space of states at intermediate stages of the calculation. 4

Applications of Quantum Computing Solving the Schrödinger Eqn. (even with fermions!) Quantum materials Machine learning* *read the fine print Quantum chemistry Cryptography and Privacy Enhancement 5

Storing information in quantum states sounds great…, but how on earth do you build a quantum computer? 6

ATOM vs CIRCUIT Superconducting circuit oscillator Hydrogen atom L C (Not to scale!) ‘Artificial atom’ 7

How to Build a Qubit with an Artificial Atom… Superconducting integrated circuits are a promising technology for scalable quantum computing T = 0. 01 K 200 nm Josephson junction: The “transistor of quantum computing” Aluminum/Al. Ox/Aluminum Al. Ox tunnel barrier Provides anharmonic energy level structure (like an atom) 8

Energy Transmon Qubit Antenna pads are capacitor plates Josephson tunnel junction ~1 mm Superconductivity gaps out single-particle excitations Quantized energy level spectrum is simpler than hydrogen Quality factor comparable to that of hydrogen 1 s-2 p Enormous transition dipole moment: ultra-strong coupling to microwave photons “Circuit QED” 9

The first electronic quantum processor (2009) was based on ‘Circuit QED’ Executed Deutsch-Josza and Grover search algorithms Lithographically produced integrated circuit with semiconductors replaced by superconductors. Michel Devoret Di. Carlo et al. , Nature 460, 240 (2009) Rob Schoelkopf 10

The huge information content of quantum superpositions comes with a price: Great sensitivity to noise perturbations and dissipation. The quantum phase of superposition states is well-defined only for a finite ‘coherence time’ Despite this sensitivity, we have made exponential progress in qubit coherence times. 11

Exponential Growth in SC Qubit Coherence “Moore’s Law” for T 2 3 D multi-mode cavity Cat Code QEC several groups 100 -200 us (Delft, IBM, MIT, Yale, …) lowest thresholds for quantum error correction NIST/IBM, Yale, . . . MIT-LL Nb Trilayer R. Schoelkopf and M. Devoret Oliver & Welander, MRS Bulletin (2013) 12

Girvin’s Law: There is no such thing as too much coherence. We need quantum error correction! 13

The Quantum Error Correction Problem I am going to give you an unknown quantum state. If you measure it, it will change randomly due to state collapse (‘back action’). If it develops an error, please fix it. Mirable dictu: It can be done! 14

Quantum Error Correction for an unknown state requires storing the quantum information non-locally in (nonclassical) correlations over multiple physical qubits. N ‘Physical’ qubits ‘Logical’ qubit Non-locality: No single physical qubit can “know” the state of the logical qubit. 15

Quantum Error Correction Cold bath N ‘Physical’ qubits ‘Logical’ qubit Entropy Maxwell Demon N qubits have errors N times faster. Maxwell demon must overcome this factor of N – and not introduce errors of its own! (or at least not uncorrectable errors) 16

All previous attempts to overcome the factor of N and reach the ‘break even’ point of QEC have failed. With major technological advances a 49 -transmon ‘surface code’ might conceivably break even at . [O’Brien et al. ar. Xiv: 1703. 04136] ‘Scale up and then error correct’ seems almost hopeless. We need a simpler and better idea. . . ‘Error correct and then scale up!’ 17

“Hardware-Efficienct Bosonic Encoding” Leghtas, Mirrahimi, et al. , PRL 111, 120501(2013). Replace ‘Logical’ qubit with this: N ‘Physical’ qubits Readout Ancilla High-Q (memory) • Cavity has longer lifetime (~ms) • Large Hilbert space • Single dominant error channel photon loss: • Single readout channel earlier ideas: Gottesman, Kitaev & Preskill, PRA 64, 012310 (2001) Chuang, Leung, Yamamoto, PRA 56, 1114 (1997) 18

Photonic Code States Can we find novel (multi-photon) code words that can store quantum information even if some photons are lost? Ancilla transmon coupled to resonator gives us universal control to make ‘any’ code word states we want. Ancilla Readout qubit High-Q (memory) 19

Quick review of microwave resonators and photonic states 20

Coherent state is closest thing to a classical sinusoidal RF signal 21

Microwave photon number distribution in a coherent state (measured via quantized light shift of qubit transition frequency) N. B. power broadened 100 X … 2 c New low-noise way to do axion dark matter detection? (ar. Xiv: 1607. 02529) Microwaves are particles!

We will use Schrödinger cat states of cavity photons (normalization is only approximate) 23

Parity of Cat States Photon number Mean photon number: 4 Even parity cat state: 8 6 4 2 0 Readout signal Coherent state: 10 Only photon numbers: 0, 2, 4, … Odd parity cat state: Only photon numbers: 1, 3, 5, … Spectroscopy frequency (GHz) Schoelkopf Lab 24

Using Schrödinger cat states to store and correct quantum information 26

Encode information in two orthogonal even-parity code “words” code word Wigner functions: + = Store a qubit as a superposition of two cats of same parity Photon loss flips the parity which (as we will see) is the error syndrome we can measure (and repeat hundreds of times). 27

Coherent states are eigenstates of photon destruction operator. Effect of photon loss on code words: After loss of 4 photons cycle repeats: We can recover the state if we know: (via monitoring parity jumps)

Coherent state Odd cat Readout signal Key enabling technology: ability to make nearly ideal measurement of photon number parity Photon number (without measuring photon number!) Even cat We learn whether n is even or odd without learning the value of n. Use dispersive coupling: Measurement is 99. 8% QND. (Can be repeated hundreds of times. ) If we can measure parity, we can perform complete state tomography (measure Wigner function) 29

2016: First true Error Correction Engine that works Analog Inputs Analog Outputs MAXWELL’S DEMON A prototype quantum computer being prepared for cooling close to absolute zero. Schoelkopf-Devoret lab 30

Implementing a Full QEC System: Debugger View (This is all real, raw data. ) Ofek, et al. , Nature 536, 441– 445 (2016). 31

Process Fidelity: Uncorrected Transmon 32

System’s Best Component 33

Process Fidelity: Cats without QEC + = 34

Process Fidelity: Cats with QEC – NO POST-SELECTION. 35

Only High-Confidence Trajectories Exclude results with heralded errors 36

Experiment: ‘Extending the lifetime of a quantum bit with error correction in superconducting circuits, ’ Ofek, et al. , Nature 536, 441– 445 (2016). Theory: ‘cat codes’ Leghtas, Mirrahimi, et al. , PRL 111, 120501(2013). ‘kitten codes’ M. Michael et al. , Phys. Rev. X 6, 031006 (2016). ‘New class of error correction codes for a bosonic mode’

QUANTUM ERROR CORRECTION Keeping your Cat Alive Courtesy of Mitra Farmand

Extra Slides 39

We can recover the state if we know: (via monitoring parity jumps) Amplitude damping is deterministic (independent of the number of parity jumps!) Maxwell Demon takes this into account ‘in software. ’

We are on the way! “Age of Qu. Error Correction. ” “Age of Quantum Feedback” “Age of Measurement” “Age of Entanglement” “Age of Coherence” Achieved goal of reaching “break-even” point for error correction. Now need to surpass by 10 x or more. M. Devoret and RS, Science (2013) 41

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Cat in Two Boxes Qubit measures joint parity! (two-legged cat only) Theoretical proposal by Paris group: Eur. Phys. J. D 32, 233– 239 (2005) 43

Cat in Two Boxes Experiment by Yale group: Science 352, 1087 (2016) Qubit measures joint parity! - Universal controllability - 3 -level qubit can measure 44

Theory Entanglement of two logical cat states 9 sigma violation of Bell inequality Experiment Two-cavities: 4 -dimensional phase space and Wigner functions. 45

Entanglement of Two Logical Cat-Qubits CHSH: (Milman et al. : evaluate Wigner at 4 points in 4 D phase space) 46

Encoding qubits in cavity photon states Minimal encoding cannot correct errors but has minimal loss rate: Minimal ‘binomial code’ corrects 1 loss: General binomial code corrects L losses, G gains and D dephasing events: M. Michael et al. , Phys. Rev. X 6, 031006 (2016) ‘New class of error correction codes for a bosonic mode’ 47

Future work: Cat pumping, non-linear driving and damping 4 -photon drive Stabilize a manifold of 4 coherent states to keep cats alive! Use parity monitoring to correct for single photon loss. 49

Photon number parity Remarkably easy to measure using our quantum engineering toolbox and Measurement is 99. 8% QND 50

Measuring Photon Number Parity - use quantized light shift of qubit frequency Gleyzes, S. et al. Nature 446, 297 (2007) Sun, Petrenko et al. , Nature 511, 444 (2014) 51

Previous Results: Tracking Photon Number Parity Jumps Parity +1 -1 odd even Syndrome msmt time: As photons leave one by one, the sign of the parity Map: ~ 200 ns changes between +1 and -1 Detect: ~ 500 ns Fidelity: > 98%, QND ~99. 8% Can perform 100’s of QND syndrome measurements per jump time Sun, Petrenko et al. , Nature 511, 444 (2014).

Using photon number parity to do cavity state tomography 53

Wigner Function = “Displaced Parity” Vlastakis, Kirchmair, et al. , Science (2013) Full state tomography on large dim. Hilbert space can be done very simply over a single input-output wire. Simple Recipe: 1. Apply microwave tone to displace oscillator in phase space. 2. Measure mean parity. Handy identity (Luterbach and Davidovitch): 54

(normalization approx. only) How cats die: 55

Transmon Qubit in 3 D Cavity Josephson junction ~ mm 50 mm Huge dipole moment: strong coupling sp in Spin flip g 100 MHz 56

Information is Physical Store Information in Quantum States? State “ 0” Quantum bit: single atom? State “ 1” Quantum superposition Principle: a quantum bit can be “ 0” and “ 1” at the same time, 57

Quantum Computing is a New Paradigm • Quantum computing: a completely new way to store and process information. • Superposition: each quantum bit can be BOTH a zero and a one. • Entanglement: the quantum computer can explore ALL possible outcomes. (Einstein: ‘Spooky action at a distance. ’) • Massive parallelism: computations that are impossible Daily routine engineering and calibration test: on ANY conventional computer. Carry out spooky operations that Einstein said were impossible! 58

The Hardware 59

QEC Setup: 2 Cavities + 1 Transmon Ancilla Input Storage Transmon JPC Readout Input Ancilla (transmon) readout fidelity ~ 99. 5% in 400 ns Output To outside Thank you 60 QLab

The Experiment: Implementing the cat code with superconducting circuits 1. Map qubit state into photonic cat code words 2. Monitor number parity jumps M 3. Conditioned on M, map cavity state back to qubit 4. Perform process tomography to determine fidelity 5. Compare error-corrected logical qubit performance to ‘best component’: (0, 1 photon encoding vs. cats) (Many slides courtesy R. Schoelkopf) 61

Quantum optics at the single photon level Large dipole coupling of transmon qubit to cavity permits: • Quantum engineer’s toolbox to make arbitrary states: ‘Dispersive’ Hamiltonian: qubit detuned from cavity -qubit can only virtually absorb/emit photons (DOUBLY QND) resonator qubit Dispersive coupling 62

Dispersive Hamiltonian resonator qubit dispersive coupling ‘strong-dispersive’ limit 63

Strong-Dispersive Limit yields a powerful toolbox Cavity frequency depends on qubit state Microwave pulse at this frequency excites cavity only if qubit is in ground state Microwave pulse at this frequency excites cavity only if qubit is in excited state Engineer’s tool #1: Conditional displacement of cavity 64

Photon number parity Remarkably easy to measure using strong-dispersive interaction between qubit and cavity (cavity and qubit linewidths) 65

Photon number parity Remarkably easy to measure using strong-dispersive interaction between qubit and cavity (cavity and qubit linewidths) Measurement is 99. 8% QND. (Can be repeated hundreds of times. ) If we can measure parity, we can perform complete state tomography (measure Wigner function) 66

Quantum Information Science Control theory, Coding theory, Computational Complexity theory, Networks, Systems, Information theory Ultra-high-speed digital, analog, microwave electronics, FPGA Programming languages, Compilers, Quantum algorithms Electrical Engineering Computer Science QIS Physics Quantum Chemistry Applied Physics Quantum mechanics, Quantum optics, Circuit QED, Materials Science quantuminstitute. yale. edu 67

Quantum Error Correction Cold bath N ‘Physical’ qubits ‘Logical’ qubit Entropy Maxwell Demon QEC is an emergent collective phenomenon: adding N-1 worse qubits to the 1 best qubit gives an improvement! 68