SIMULATING AND CORRECTING QUBIT LEAKAGE Martin Suchara Andrew

SIMULATING AND CORRECTING QUBIT LEAKAGE Martin Suchara Andrew Cross Jay Gambetta Supported by ARO W 911 NF-14 -1 -0124 September 12, 2014

What is Qubit Leakage? o Physical qubits are not ideal two-level systems and may leak out of the computational space Leakage Bit flip o With standard error correction techniques leaked qubits accumulate and spread errors o This talk: simple model of leakage and comparisons of leakage reduction strategies 2

Leakage in the Literature o First mentions of leakage detection (Gottesman, 1997, Preskill 1998) o Analysis of leakage reduction units based on quantum teleportation, threshold theorem for concatenated codes (Aliferis, Terhal, 2005) o Model of leakage for repetition code that labels leaked qubits (no quantum simulation) (Fowler, 2013) 3

Overview I. Our leakage model II. A few examples of leakage reduction circuits III. Error decoding strategies IV. Thresholds and error rates with leakage reduction 4

Abstract Model of Leakage: Erasures o Leakage event is probabilistic erasure of qubit o Leaked qubits may decay back to qubit space 5

How Should Two-Qubit Gates Behave? o Assume gates are direct sums of unitaries o Assume unitaries of m 2 and 2 m blocks are maximally entangling and twirl over the L subsystem on these blocks after each gate o If only one input leaks, this is equivalent to depolarizing the unleaked input Model acts violently on the leakage subspace between gates. 6

Simulating Leakage for the Toric Code Z Z X X Data qubit Ancilla Stabilizers o Our label-based model: each qubit is in state I, X, Y, Z, or L 7

Simulating by Propagating Labels o Our leakage model destroys syndrome correlations of less violent models E s 1 s 3 |2 s 4 s 2 parity constraints violated with probability ½ since ancilla depolarized o Does not appear necessary to retain quantum state in the simulation - conjecture propagating new error label faithfully simulates the model for surface code 8

Behavior of Gates Gate Identity Preparation Measurement Possible Errors Leakage Errors X, Y, Z if leaked relaxes w/ prob. pd, doesn’t increase leakage orthogonal state leaks w/ prob. pu incorrect if leaked, always measures 1 (also consider leakage detection) IX, XZ, etc. if leaked, applies random Pauli to the other qubit; leaks w/ prob. pu and relaxes w/ prob. pd CNOT 9

C++ Simulation Measures and Matches Error Syndromes X X Z o Use minimum weight matching and correct errors between pairs of closest syndromes o Circuit model simulates syndrome errors 10

Circuit Model of Syndrome Extraction d. D d. R d. L d. U a. X s a. Z s o Each gate in the circuit causes Pauli errors or leakage according to our model 11

Leakage can Accumulate o Leakage accumulates on the data qubits o Initialization of ancillas prevents accumulation o Equilibrium leakage rate is a property of the circuit and its gates n Our circuit: n 4 pu: leakage caused by CNOTs n 6 pd: leakage reduction of CNOTs and identities 12

Simulation Details o Start simulation in equilibrium n A fraction of data qubits starts in L state o A round of perfect leakage reduction at the end of each simulation n Leaked qubit replaced with I, X, Y, or Z o We use d rounds of syndrome measurements, the last one is ideal 13

Success Probabilities No threshold pth ~ 0. 66% o Leakage reduction is necessary! Only works for p = 0. 02% 14

Overview I. Our leakage model II. A few examples of leakage reduction circuits III. Error decoding strategies IV. Thresholds and error rates with leakage reduction 15

Full-LRU Circuit d 1 d 1 d 2 d 2 o Swap with a newly initialized qubit after each gate o Slow and expensive 16

Partial-LRU Circuit a 4 d. D d. U d. L d. R d. D a. Z s a. X d. D s o Swaps each data qubit with a fresh one during ancilla measurement o Requires 3 CNOTs 17

Quick Leakage Reduction Circuit d. U d. L d. R d. D a. Z d. D s d. D a. X s d. D o Swaps data qubits and ancillas o Sufficient to add a single CNOT gate 18

Overview I. Our leakage model II. A few examples of leakage reduction circuits III. Error decoding strategies IV. Thresholds and error rates with leakage reduction 19

The Standard and Heralded Leakage (HL) Decoders o Standard Decoder only relies on syndrome history to decode errors o HL Decoder uses leakage detection when qubits are measured o Partial information about leakage locations o Error decoder must be modified 20

Standard Decoder for the Toric Code o Need to correct error chains between pairs of syndromes o Decoding graphs for X and Z errors built up using this unit cell (Fowler 2011) o Need to adjust edge weights for each leakage suppressing circuit (Full-LRU, Partial-LRU, Quick circuit) 21

Standard Decoder – Adjustment of Edge Weights Circuit a No-LRU 11/5 p + q 28/15 p 16/15 p 7/3 p + q 32/15 p Quick circuit Full-LRU Partial-LRU b c d e f 52/15 p 8/15 p 4 p 8/15 p 32/15 p 103/15 p + q 52/15 p 88/15 p 172/15 p 32/15 p 8/15 p 31/15 p + q 8/15 p 4/3 p 52/15 p 16/15 p 76/15 p 22

HL Decoder: Quick Circuit (11 leakage locations) 23

HL Decoder: Partial-LRU Circuit (5 ancilla leakage locations) 24

HL Decoder: Partial-LRU Circuit (9 data leakage locations) 25

Overview I. Our leakage model II. A few examples of leakage reduction circuits III. Error decoding strategies IV. Thresholds and error rates with leakage reduction 26

Threshold Comparison o More complicated circuits have lower threshold o HL decoder helps boost the threshold 27

Decoding Failure Rates o Full-LRU performs well at low error rates 28

Effect of the Leakage Relaxation Rate (Quick circuit) o Leakage relaxation rate small compared to the leakage suppression capability of the circuits 29

Conclusion o Leakage reduction is necessary o Model of leakage that allows efficient simulation o A simple leakage reduction circuit that only adds a single CNOT gate and new decoders o Systematic exploration of error correction performance o Available as ar. Xiv 1410. 8562 30

Thank You! 31