Download presentation

Presentation is loading. Please wait.

Published byMegan Patterson Modified over 2 years ago

1
30 August 2005 Christopher Dawson Henry Haselgrove Michael Nielsen Noise thresholds for optical quantum computers

2
n To numerically find the noise threshold for cluster-state linear optical quantum computing (LOQC) u Thereby, help judge feasibility of implementing LOQC. n Our final result will be a noise threshold curve. u We imagine that each optical element is subject to depolarization noise and photon loss noise. Introduction: our aim u We determine the range of these noise strengths for which fault tolerant error correction can reduce the error rate to zero: (below the threshold) (above the threshold) The threshold Depolarization rate (per optical element) Photon loss rate (per optical element)

3
Background n Linear-optical quantum computing (LOQC) u Qubit: encoded in polarization of a single photon u Resources: Single-photon sources, passive linear optics (beam splitters, phase delays, wave plates), photon-counting photodetectors. u Major advantage: photons can be isolated from environment u Major disadvantage: photon-photon interactions difficult n Knill, Laflamme, & Milburn: u Devised the nondeterministic controlled phase (CPHASE) gate u Fundamentally nondeterministic (not due to noise) Prob. Success ¼ 1/20 |v i control target

4
n Difficulty of KLM u CPHASE becomes extremely complex (1000s of optical elements) when probability of success is made high n Examples of improved LOQC schemes u Nielsen: F Computation is performed in the cluster-state model. F The cluster state can be built efficiently using the low-success- probability (i.e. simple) version of KLM CPHASE gate. F Overall optical circuit is simplified as a result u Browne and Rudolf: F Also uses cluster-state model, using even simpler fusion gate as alternative to CSIGN. F Results in further simplification to the optical circuit n The scheme we simulate : u Takes elements of Nielsen, and Browne and Rudolf u Plus further modifications Improvements to KLM

5
Related threshold results n A range of threshold estimates (numerical and analytical) have been performed before. u For example, Steanes comprehensive numerical threshold simulations (Circuit model, and simple depolarization noise) n Such results dont directly apply to the situation we consider u Our protocol operates in the cluster-state model, not the circuit model u Our noise model is necessarily more complicated (two noise types, having very different effects) n Analytical results for cluster state model: u Nielsen and Dawson showed that a threshold exists in the cluster- state model u Simplified argument by Aliferis & Leung u These proofs give a bound on the threshold, but not a precise value

6
Physical setting n Resources: u Source of Bell pairs (simple two-node cluster state) F Perhaps generated using parametric downconversion u Photon-number discriminating photodetectors u Passive linear optics u Quantum memory n Qubits, and operations on qubits Dual-rail qubits: |0 i + |1 i ! |H i + |V i u Single-qubit measurements and gates very simple combinations of above resources u The fusion gate (to build cluster states, described later) n Error model: u Each qubit operation (fusion, memory, Bell preparation, measurement) has a possibility of introducing depolarisation and/or photon loss. u Nondeterminism of fusion gates: they fail with probability 1/2. u For convenience, no dark counts (false positive photon counts)

7
Remainder of the talk n A little more background: u Cluster state model u Fusion gate u Building cluster states optically with fusion gate n Our cluster-based error-correction protocol n The simulation, and final threshold results

8
Cluster state computing n Raussendorf and Briegel: u Measuring each qubit of a cluster state is universal for quantum computing. u That is, any quantum circuit can be simulated by first creating, then measuring, a cluster state. n Cluster states: u most general notion, often called graph state u For every graph, there is a corresponding cluster state. For example: |+ i )

9
Cluster state computing n Converting a quantum circuit to a cluster state computation: u Write the circuit in terms of the universal set: F Controlled phase two-qubit gate F He -i Z == HZ (family of single qubit gates) u Replace each HZ in the circuit as follows: HZ x X |+ i

10
Example conversion HZ |+ i HZ x x X X |+ i HZ x x Cluster creation Measurement Classical feed-forward

11
The fusion gate Behaviour of the gate depends on how many photons are detected: n Success: Defined to be when the photodetector counts exactly one photon. Then, output relates to the input by the operator |0 ih 00| + |1 ih 11| n Failure: Defined to be when the photodetector counts zero or two photons. Then, computational basis measurement is performed on input qubits |01 i and |10 i. No qubit is output. input qubit 1 input qubit 2 45° output qubit (Polarization-discriminating photodetector) n The fusion gate has two inputs and one output

12
Building cluster states optically Effect of a fusion gate on a cluster state depends on the success or failure of the gate n Successful fusion gate: combines two nodes of a cluster ) n Failed fusion gate: removes nodes from cluster ) In our protocol, fusion gates build clusters from Bell pairs. The cluster is equivalent to a Bell pair. (50% probability) (50% probability)

13
How do you build up clusters efficiently with fusion gates that fail 50% of the time? n First build a supply of microclusters by fusing Bell states n Use microclusters as building blocks. To join with many parallel attempts at fusion n Chance of join succeeding can be made arbitrarily high Fusion with higher probability of success (creating a k-leaf microcluster takes on average k 2 Bell pairs)

14
Clusterized error correction protocol n Similar elements present, in a disguised form, in the cluster-state protocol. data A A A A F.T. ancilla creation X syndrome extractions Z syndrome extractions For comparison, traditional fault-tolerant QEC:

15
Clusterized protocol data, plus dangling nodes A A A A Cluster for data-ancilla interaction data AA A A (equivalent circuit) X synd. Z synd. X synd. ancilla cluster

16
Ancilla creation j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i H H H H

17
j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i H H H H

18
n The above cluster is created by fusing microclusters n Then all qubits in columns 1 to 7 are measured, to run the cluster u (leaves column 8 so we can join main cluster) n Verification bits (output of first four rows) are checked n Additional verification check: no lost photons

19
n Verified ancillas are joined to form the telecorrector cluster (this cluster both teleports and corrects the data) Pre-running the telecorrector: n Even before we interact (join) the telecorrector with the data, we do the following: u Measure all the dark-coloured qubits, to pre-run part of the cluster u (This pre-running commutes with the process of joining the telecorrector to the data) telecorrector cluster input data cluster A A A A

20
Advantages of pre-running the telecorrector qubits: We can check for a range of different types of errors on the telecorrector, and throw it away if necessary: u Disagreeing syndromes. F Normally, disagreeing syndromes cause a QEC round to be wasted, just adding more noise to data. F We can throw away telecorrectors with disagreeing syndromes. Effect will be to improve threshold. u Lost photons. F When the telecorrector is pre-measured, lost photons are easily detected. Thus, lost photons on these qubits dont add noise to data u Nondeterminism of fusion gates. F We only use a telecorrector when we know the construction of it has succeeded.

21
n When we have a verified telecorrector, we attach it to data, and measure data to finish running the cluster. n Many fusion-gate attempts per row needed. Multiple fusion gate attempts, followed by measurement Error-corrected data n Photon loss and nondeterminism at this stage: u effects output data, but we know which row. F These are located errors. F Decoding routine takes advantage of this knowledge u Dont need to replace a lost photon: qubit being teleported onto almost certainly still has a photon.

22
n We perform a many-trial Monte Carlo simulation u Stochastically introduce errors according to noise model u Track errors as they propagate through the circuit n Measure the resulting rate of Pauli errors on the encoded qubit, that is crashes n Two very different types of crashes: u Located crashes: - The experimenter knows that the encoded state has suffered depolarization. Triggered when many qubits in the code experience located errors, e.g. photon loss. u Unlocated crashes: - Crashes not known to the experimenter. Mainly caused by combinations of depolarization errors. How we simulate the protocol (photon loss rate, depolarization rate) ! (loc. crash rate, unloc. crash rate) Input parameters ! Output statistics

23
n How do you know when a simulation of a fault-tolerant computation is working bug-free? Can results be verified? n Our approach: Write two versions of the same simulator independently, and compare results! n Look for bugs until simulators agree completely. Our simulator: a redundancy code of sorts

24
n Noise levels are below the threshold when repeated concatenation of error-correction protocols reduce the effective error rate to zero Thresholds and concatenation Photon loss rate Depolarization rate Loc. crash rate = loc. error rate unloc. crash rate = unloc. error rate Level 1 (cluster protocol) Level 2 (circuit-based protocol) Loc. crash rate… unloc. crash rate …= loc. error rate …= unloc. error rate Level 3 (circuit-based protocol) …etc.

25
Deterministic (circuit-based) protocol n Used for second and higher levels of concatenation n Inspired by cluster model, this protocol also uses a telecorrector u (Syndromes are extracted before any interaction with data) j0ij0ij0ij0i j0ij0i j0ij0i j+inj+in j+inj+in Data (telecorrector creation) Schematic showing the order of syndrome extractions in our circuit- based telecorrection protocol: n Noise model: unlocated and located errors. Legend (measurement types): teleportation Z syndrome extraction X syndrome extraction

26
Circuit-model results: flow diagram x located error rate, q unlocated error rate, p (using 23-qubit Golay code)

27
Polynomial fitting. Deterministic threshold x located error rate, q unlocated error rate, p (using 23-qubit Golay code)

28
Final threshold results n We fit polynomials to the map (input noise rates) (crash rates) for both protocols u 1. Optical protocol u 2. Circuit-based protocol n Can then test any value for the physical noise rates, very quickly, by applying map 1 once and map 2 many times n Result: high-resolution threshold curve with respect to the physical noise rates n Carried out whole procedure for two code types: u 7-qubit Steane code u 23-qubit Golay code

29
Final threshold results 7-qubit code, memory noise disabled23-qubit code, memory noise disabled 7-qubit code, all noise types enabled 23-qubit code, all noise types enabled x Photon loss rate, Depolarization parameter, x Photon loss rate, Depolarization parameter, x x Photon loss rate, Depolarization parameter, x x Photon loss rate, Depolarization parameter,

30
x Photon loss rate, Depolarization parameter, Final threshold results 7-qubit code, memory noise disabled23-qubit code, memory noise disabled 7-qubit code, all noise types enabled 23-qubit code, all noise types enabled x Photon loss rate, Depolarization parameter, x x Photon loss rate, Depolarization parameter, x x Photon loss rate, Depolarization parameter,

31
Conclusions n In principle, reliable LOQC can be performed with combined error rates, per physical operation, of: u photon loss rate Pauli error rate 2 £ Threshold is worse than circuit-model threshold (as it should be: nondeterministic gates). Not too much worse though. n Using the cluster state model in linear optics quantum has several advantages u Use of the simple fusion gate as building block u Advantages associated with the teleported nature of cluster state computing F Post-selection for pre-agreeing syndromes F Post-selection against located noise types

32

33
* (using this noise rate) Example of rough resource-usage calculation Level# Bell pairs per EC round Unloc. crash rate Loc. crash rate 11 × × × × × × × × × × × × × × ×

34
Polynomial fitting. Deterministic threshold x Located error rate, q Unlocated error rate, p (using 23-qubit Golay code) (note: protocol performs better with small amounts of located noise!)

35
j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i H H H H

36
j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i H H H H

37
j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i H H H H

38
j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i H H H H

39
j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i H H H H

40
j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i j+ij+i H H H H

Similar presentations

© 2016 SlidePlayer.com Inc.

All rights reserved.

Ads by Google