Quantum computers still work with 25% of their qubits missing

Transcription

1 Quantum computers still work with 25% of their qubits missing Sean Barrett Tom Stace (UQ) Phys. Rev. Lett. 102, (2009) Phys. Rev. A 81, (2010) Phys. Rev. Lett. 105, (2010)

2 Outline Loss/leakage errors are a serious problem for practical QIP This motivates development of error correction / fault tolerance schemes tailored for loss errors Kitaev s surface codes are robust to loss errors, with a very high threshold (percolation) Raussendorf s topological quantum computer is robust to computational and loss errors, with high thresholds for both pcomp Correctable Uncorrectable p loss

3 Loss errors in QIP Loss errors can be a serious (dominant) source of noise in various QIP architectures Photonic QIP: mode mismatch, imperfect sources, inefficient detectors Trapped atoms/ions: imperfect loading of lattices, leakage Solid state architectures: fabrication errors Greiner et al, Nature 415, 39 (2002)

4 Toric code I Qubits live on edges of L x L lattice

5 Toric code I Qubits live on edges of L x L lattice Two types of stabilizer operator: stars and plaquettes

6 Toric code I Qubits live on edges of L x L lattice Two types of stabilizer operator: stars and plaquettes Disjoint stabilizers (obviously) commute Overlapping stabilizers always share two qubits [P i,s j ]=0

7 Toric code I One of the stars (plaquettes) may be expressed as product of all others So generators of stabilizer are: S = P 1,P 2,..., P L 2 1,S 1,S 2,..., S L 2 1 Number of encoded qubits is thus k =2L 2 2(L 2 1) = 2

8 Toric code II What are the logical operators? First, consider effect of a chain of operators: An open chain commutes with S except at endpoints Closed chains ( cycles ) commute with S Homologically trivial cycle member of S Homologically non-trivial cycle logical operator

9 Correction procedure Measure all P 1,P 2,..., P L 2 1,S 1,S 2,..., S L 2 1 Reveal endpoints of error chains For error chains E, need to find correction chains E such that E+E is homologically trivial This can be done with a minimum weight matching algorithm Find p c = 10.3%

10 Loss errors Consider effect of losses in absence of Pauli errors

11 Loss errors Consider effect of losses in absence of Pauli errors When we loose qubits, can reroute logical operators: ' = P i C = P i C = C

12 Loss errors Consider effect of losses in absence of Pauli errors When we loose qubits, can reroute logical operators: ' This will work provided threshold p loss is less than percolation Relevant number is square lattice bond percolation threshold: p loss < 0.5

13 QEC for loss & Pauli errors When qubits are lost, plaquettes (and stars) can no longer be measured unambiguously??

14 QEC for loss & Pauli errors When qubits are lost, plaquettes (and stars) can no longer be measured unambiguously?? Solution: multiply damaged operators to get new stabilizers which can be measured unambiguously

15 QEC for loss & Pauli errors Algorithm: merge nodes on stabilizer graph Do perfect matching on reduced graph

16 QEC for loss & Pauli errors Algorithm: merge nodes on stabilizer graph Do perfect matching on reduced graph Results: Uncorrectable pt Correctable 0.02 Phys. Rev. Lett. 102, (2009) Phys. Rev. A 81, (2010) p loss

17 Topological FTQC scheme Raussendorf, Harrington & coworkers introduced a FTQC scheme inspired by topological quantum computing Measurement based QC Translationally invariant Nearest neighbour gates only Works in two dimensions

18 Topological FTQC scheme Raussendorf, Harrington & coworkers introduced a FTQC scheme inspired by topological quantum computing Measurement based QC Translationally invariant Nearest neighbour gates only Works in two dimensions p thres =

19 Topological FTQC scheme MBQC: start with cluster state on a 3D lattice, L Qubits live on faces & edges of L

20 Topological FTQC scheme MBQC: start with cluster state on a 3D lattice, L C gates Qubits live on faces & edges of L

21 Topological FTQC scheme MBQC: start with cluster state on a 3D lattice, L C gates Qubits live on faces & edges of L Divide qubits into three types: D V S qubits, measured in qubits, measured in qubits, measured in Y or ( + Y )

22 Topological FTQC scheme MBQC: start with cluster state on a 3D lattice, L C gates Qubits live on faces & edges of L Divide qubits into three types: D V S qubits, measured in qubits, measured in qubits, measured in Clifford gates Y or ( + Y ) Universality (via magic state purification)

23 Consider simplest encoded gate: identity gate Identity gate

24 Consider simplest encoded gate: identity gate Identity gate Defect regions

25 Consider simplest encoded gate: identity gate Identity gate Defect regions Vacuum

26 Consider simplest encoded gate: identity gate Identity gate Defect regions Logical qubits encoded in surface code Vacuum

27 To understand how this works (even without errors), make use of stabilizers K i C L = C L, Identity gate

28 To understand how this works (even without errors), make use of stabilizers K i C L = C L, Identity gate K i =

29 To understand how this works (even without errors), make use of stabilizers K i = Products of form: K i C L = C L, K i Identity gate s have an intuitive

30 To understand how this works (even without errors), make use of stabilizers K i = Products of form: K i C L = C L, K i s have an intuitive Identity gate Correlation surfaces

31 By considering these correlation surfaces, we can show: ± in out C L = C L, ± in out C L = C L, Identity gate

32 By considering these correlation surfaces, we can show: ± in out C L = C L, ± in out C L = C L, Identity gate After measuring input slice, are mapped to and out out in and in

33 By considering these correlation surfaces, we can show: ± in out C L = C L, ± in out C L = C L, Identity gate After measuring input slice, are mapped to and out out in and in Input state has been teleported to output plane

34 Gates with errors Now consider the effect of errors Consider products of K i s around a cube of the lattice

35 Gates with errors Now consider the effect of errors Consider products of K i s around a cube of the lattice

36 Gates with errors Now consider the effect of errors Consider products of K i s around a cube of the lattice Cubes with product -1 reveal the locations of endpoints of error chains

37 Gates with errors Now consider the effect of errors Consider products of K i s around a cube of the lattice Cubes with product -1 reveal the locations of endpoints of error chains Minimum weight matching problem

38 Loss errors First idea: correlation surfaces can by deformed by multiplying with {K i } Deform the surfaces so that they avoid lost qubits

39 Loss errors First idea: correlation surfaces can by deformed by multiplying with {K i } Deform the surfaces so that they avoid lost qubits

40 Loss errors First idea: correlation surfaces can by deformed by multiplying with {K i } Deform the surfaces so that they avoid lost qubits As long as we don t loose too many qubits, we are able to reroute all the correlation surfaces and the gate still works

41 Thresholds Rerouting the correlation surfaces is dual to the problem of bond percolation on the 3D lattice. We expect that failure threshold coincides with bond percolation threshold in 3D.

42 Thresholds Rerouting the correlation surfaces is dual to the problem of bond percolation on the 3D lattice. We expect that failure threshold coincides with bond percolation threshold in 3D p c correctable p loss 0.248

43 Loss errors In the presence of computational errors and loss errors, can still use parity checks to detect computational errors

44 Loss errors In the presence of computational errors and loss errors, can still use parity checks to detect computational errors Idea: join parity check cubes together to avoid lost faces

45 Loss errors In the presence of computational errors and loss errors, can still use parity checks to detect computational errors Idea: join parity check cubes together to avoid lost faces C E C C c E E b Minimum weight matching on modified graph

46 Results We performed Monte-Carlo simulations of error & recovery processes on finite size lattices (L = 8x8x8,...,16x16x16) Error model: computational errors in preparation, storage, C gates, measurements, with equal rate p comp Loss errors with rate p loss pcomp Correctable Uncorrectable Phys. Rev. Lett. 105, (2010) p loss

47 Conclusions We have developed methods for overcoming loss errors in fault tolerant quantum computing A modification of Raussendorf s scheme for FTQC is extremely robust to both computational and loss error pcomp Correctable Uncorrectable Ongoing work: nondeterministic gates (e.g. linear optics), static defects, overhead costs... Herrera-Martí et al., Phys. Rev. A 82, (2010) Li et al. Phys. Rev. Lett. 105, (2010) (talk tomorrow) Fujii and Tokunaga, Phys. Rev. Lett. 105, (2010) (poster) Nagayoma et al. (poster) p loss