Measurement-based Quantum Computation

Transcription

1 Measurement-based Quantum Computation Elham Kashefi University of Edinburgh

2 Measurement-based QC Measurements play a central role. Scalable implementation Clear separation between classical and quantum parts of computation Entanglement Clear separation between creation and consumption of resources

3 Basic Commands New qubits, to prepare the auxiliary qubits: N Entanglements, to build the quantum channel: E Measurements, to propagate(manipulate) qubits: M Corrections, to make the computation deterministic: C

4 2-state System C 2 The canonical basis, (1, 0), (0, 1), also called the computational basis, is usually denoted 0, 1. It is orthonormal by definition of x, y C 2. ± := 1 2 ( 0 ± 1) ± α := 1 2 ( 0 ± e iα 1) The preparation map N α i is defined to be: + α : H n C 2 H n

5 Maps over C 2 Pauli Spin Matrices X := Z := Other Single qubit gates H := P (α) := e iα P (α) = P ( α).

6 The two qubit state C 2 C 2 Canonical basis { 00, 01, 10, 11} Bases need not be made of decomposable elements, they can consists of entangled states. G Graph basis G 00 = 1 2 ( ) G 01 = 1 2 ( ) G 10 = 1 2 ( ) G 11 = 1 2 ( )

7 Maps on C 2 C 2 In general if f : A B and g : A B, one defines f g : A A B B : ψ φ f(ψ) g(φ). Or given f : C 2 C 2, one defines f (read controlled-f) a new map on C 2 C 2 : f 0 ψ := 0 ψ f 1 ψ := 1f( ψ) Entangling Map Z( + +) =G 00

8 Pauli and Clifford Define the Pauli group over A as the closure of {X i,z i 1 i n} under composition and. These are all local maps (corrections). Define the Clifford group over A as the normalizer of the Pauli group, that is to say the set of unitaries f over A such that for all g in the Pauli group, fgf 1 is also in the Pauli group. Entangling Map is in Clifford Z ij X i = X i Z j Z ij Z ij Z i = Z i Z ij

9 Projective Measurement on H n A complete measurement is given esby aan orthonormal basis B = {ψ a } thogonal ( which defines a decomposition into orthogonal 1-dimensional subspaces H n = a E a Define ψ a ψ a : H n E a to be projection to Ea, Outcome M B : H n a E a : φ a ψ a, φ ψ a Probability

10 Destructive Measurement Given a complete measurement over A, as A = {ψ a }, one can extend it to an incomplete measurement on A B, with components given by ψ a ψ a : A B B. 1-qubit destructive measurement M α { associated to M for t { + α }.

11 Unitary Action If U maps orthonormal basis B to A then: B A M A = UM B U X-action: X + α = + α X α = α Z-action: Z + α = + α+π Z α = α+π

12 Quantum Pacman

13 Quantum Pacman

14 Quantum Pacman

15 Quantum Pacman

16 A formal language N i prepares qubit in + = 1 2 ( 0 + 1) = M α i projects qubit onto basis states (measurement outcome is s i = 0, 1 ) E ij creates entanglement ± α = 1 2 ( 0 ± e iα 1) = ±e iα Local Pauli corrections X i = , Z i = Feed forward: measurements and corrections commands are allowed to depend on previous measurements outcomes. i i Ci s [Mi α]s = M ( 1)s α i s[mi α] = M i α+sπ +

17 Dependent Commands The measurement outcome s i Z 2 : 0 refers to the + α projection, 1 refers to the α projection. measurements and corrections may be parameterised by signal i s i [Mi α]s = M ( 1)s α i t [Mi α ]=M tπ+α i = M α i Xs i = M α i Zs i with X 0 = Z 0 = I, X 1 = X, Z 1 = Z. t[m α i ]s = M tπ+( 1)s α i

18 Patterns of Computation (V, I, O, A n... A 1 ) H := ({1, 2}, {1}, {2}, X s 1 2 M 0 1 E 12N 0 2 ) Sequential or Parallel Composition X s 2 3 M 0 2 E 23 X s 1 2 M 0 1 E 12

19 Definiteness Conditions no command depends on outcomes not yet measured no command acts on a qubit already measured a qubit i is measured if and only if i is not an output

20 Example H := ({1, 2}, {1}, {2}, X s 1 2 M 0 1 E 12N 0 2 ) Starting with the input state (a 0 + b 1) + 0 (a 0 + b 1) + 1 (a 00 + a 01 + b 10 b 11) 2 H E 12 M ((a + b) 0 + (a b) 1) s 1 = ((a b) 0 + (a + b) 1) s 1 = 1 X s ((a + b) 0 + (a b) 1)

21 State Space S := V,W H V Z W 2 In other words a computation state is a pair q, Γ, where q is a quantum state and Γ is a map from some W to the outcome space Z 2. We call this classical component Γ an outcome map and denote by the unique map in Z 2.

22 Operational Semantics q, Γ N α i q, Γ E ij q, Γ Xs i q, Γ Zs i q, Γ t [Mi α ]s q, Γ t [M α i ]s q + α i, Γ Z ij q, Γ X s Γ i q, Γ Z s Γ i q, Γ + αγ i q, Γ[0/i] αγ i q, Γ[1/i] where α Γ = ( 1) s Γα + t Γ π.

23 Denotational Semantics H I H I Z 2 prep H V Z 2 H O H O Z V O 2 Let A s = C s Π s U be a branch map, the pattern realises the cptp-map T (ρ) := s A s ρa s Density operator : A probability distribution over quantum states

24 Denotational Semantics H I H I Z 2 prep H V Z 2 H O H O Z V O 2 A pattern is strongly deterministic if all the branch maps are equal. Theorem. A strongly determinist pattern realises a unitary embedding.

25 Universal Gates Z := J(α) := e iα 1 e iα U = e iα J(0)J(β)J(γ)J(δ) P (α) = J(0)J(α) H = J(0) H i = J( π 2 )

26 Generating Patterns Z := Z := E 12 J(α) := e iα 1 e iα J(α) := X s 1 2 M α 1 E 12

27 Example (ctrl-u) U = e iα J(0)J(β)J(γ)J(δ) U 12 = J1 0Jα 1 J0 2 Jβ+π 2 J γ 2 2 J π 2 where: α = α + β+γ+δ 2 2 J0 2 Z 12J π γ π δ β 2 2 J 2 2 J 2 2 J 0 2 Z 12J β+δ π 2 2

28 Example (ctrl-u) X s B C M 0 B E BCX s A X s h i M γ 2 X s d e M γ 2 d h E hix s g E de X s c B MA α E ABX s j k M 0 j E jkx s i π h M 2g E gh X s f g Mf 0E fge Af X s e π+δ+β d M 2 c Wild Pattern j M β π i E ij f M π 2 e E ef E cd X s b c M 0 b E bce Ab X s a β δ+π b M 2 a E ab Standard Pattern Z s i+s g +s e +s c +s a k M j 0 M 0 B M α A [M β π X s j+s h +s f +s d +s b k X s B C Z s A+s e +s c C i ] s h+s f +s d +s b [M γ 2 Mf 0[M π 2 e ] s γ π δ β d+s b [M 2 d ]s c+s a [M 2 c ] s bmb 0 β+δ+π M 2 a E BC E AB E jk E ij E hi E gh E fg E Af E ef E de E cd E bc E ab E Ab h ]s g+s e +s c +s a [M π 2 g ] s f+s d +s b

29 Measurement Calculus Pushing entanglement to the beginning E ij X s i = X s i Z s j E ij E ij X s j = X s j Z s i E ij E ij Z s i = Z s i E ij Pushing correction to the end t [M α i ] s X r i = t [M α i ] s+r t [M α i ] s Z r i = t+r [M α i ] s E ij Z s j = Z s j E ij Theorem. The re-writing system is confluent and terminating. Theorem. An MQC model admits a standardisation procedure iff the E operator is normaliser of all the C operators.

30 Algorithm U = e iα J(0)J(β)J(γ)J(δ) J(0)(4, 5)J(α)(3, 4)J(β)(2, 3)J(γ)(1, 2) = X s 4 5 M 4 0E 45X s 3 4 M 3 αe 34X s 2 3 M β 2 E 23X s 1 2 M γ 1 E 12 EX X s 4 5 M 4 0E 45X s 3 4 M 3 αe 34X s 2 3 M β 2 Xs 1 2 Zs 1 3 M γ 1 E 123 MX X s 4 5 M 4 0E 45X s 3 4 M 3 αe 34X s 2 3 Z3 s 1 [M β 2 ]s 1M γ 1 E 123 EXZ X s 4 5 M 4 0E 45X s 3 4 M 3 αxs 2 3 Z3 s 1 Z s 2 4 [M β 2 ]s 1M γ 1 E 1234 MXZ X s 4 5 M 4 0E 45X s 3 4 Zs 2 4 s 1 [M 3 α]s 2[M β 2 ]s 1M γ 1 E 1234 EXZ X s 4 5 M 4 0Xs 3 4 Zs 2 4 Zs 3 5 s 1 [M 3 α]s 2[M β 2 ]s 1M γ 1 E MXZ ment com O(N 5 ). section = N C for Worst Case Complexity: where is the number of qubits in the given pattern

31 The Key Feature of MBQC A clean separation between Classical and Quantum Control A B C j a b c d e f g h i j k A B C k Entanglement Graph a b c d e g h i f Execution Graph

32 Parallelisation A B C j a b c d e f g h i j k A B C k Entanglement Graph a b c d e g h i f Execution Graph Signal Shifting t [Mi α]s Si t[m i α]s Xj sst i Si txs[t+s i/s i ] j Zj sst i Si tzs[t+s i/s i ] t j [Mj α]s Si r Si r t[r+s i/s i ] [Mj α]s[r+s i/s i ] S s i St j S t j Ss[t+s j/s j ] i

33 Reducing Depth Depth of a pattern is the length of the longest feed-forward chain Standardisation and Signal Shifting reduce depth. Z s b g X s d g Z s b f Z s a f Z s a e X s c e [Mδ d ]s b [Mc γ ] s a sa [M β b ]Mα a E G Z s b g X s d g Z s b f Z s a f Z s a e X s c e [M δ d ]s b [M γ c ]s a s a [Mβ b ] Mα a E G Z s b g X s d g Z s b f Z s a f Z s a e X s c e [M δ d ]s b S s a b [M γ c ]sa M β b Mα a E G Z s b g X s d g Z s b f S s a b Z s a f Z s a e X s c e [M δ d ]s b +sa [M γ c ]sa M β b Mα a E G Z s b g S s a b Z s b +s a g X s d g X s d g Z s b f Z s b +s a f Z s a e X s c e Z s a f Z s a e X s c e [M δ d ]s b +sa [M γ c ]sa M β b Mα a E G [M δ d ]s b +sa [M γ c ]sa M β b Mα a E G

34 Depth Complexity All the models for QC are equivalent in computational power. Theorem. There exists a logarithmic separation in depth complexity between MBQC and circuit model. Parity function: MQC needs 1 quantum layer and circuit model the quantum depth is Ω(log n) O(log n) classical layers whereas in the A. Broadbent and E. Kashefi, MBQC07

35 Automated Parallelising Scheme Theorem. Forward and backward translation between circuit model and MQC can only decrease the depth.

36 Characterisation tern P has d P N i d P d + 2 Theorem. A pattern has depth if and only if on any influencing path we obtain after applying the following rewriting rule: { NN if P N P1 α 1 β 1 P2 α 2 β 2 Pk N i N otherwise X(XY )

37 The Magical Clifford Sequence! (H) odd (H i (H) odd )!

38 Example ψ 1 α 11 H α 12 H α 1n H ψ 2 α 21 H α 22 H α 22 H ψ 3 α 31 α 32 α ψ n 1 H H H ψ n α n1 α n2 α nn Can be parallelised to a pattern with depth 2

39 Determinism A pattern is deterministic if all the branches are the same. How to obtain global determinism via local controls A necessary and sufficient condition for determinism based on geometry of entanglement

40 Graph State as Stabiliser States oof ofgraph the flow Stabilisers: theorem (T K i := X i ( j N G (i) Z j) K i E G N I c = E G N I c

41 Graph State as Stabiliser States oof ofgraph the flow Stabilisers: theorem (T K i := X i ( j N G (i) Z j) K i E G N I c = E G N I c Z X Z

42 Graph State as Stabiliser States oof ofgraph the flow Stabilisers: theorem (T K i := X i ( j N G (i) Z j) K i E G N I c = E G N I c Z Z X Z

43 Graph State as Stabiliser States oof ofgraph the flow Stabilisers: theorem (T K i := X i ( j N G (i) Z j) K i E G N I c = E G N I c

44 Flow Definition. An entanglement graph f : O c I c and a partial order (G, I, O) has flow if there exists a map over qubits (i) x f(x) (ii) x f(x) (iii) for all y f(x), we have x y

45 Flow Find a qubits to qubits assignment a matching partial order

46 Flow Find a qubits to qubits assignment a matching partial order

47 Constructive Determinism Theorem. A pattern is uniformly and step-wise deterministic iff its graph has a flow. 4 2 Z s Z s1 X s1 3 Z s1 i O c(xs i f(i) k N G (f(i)){i} Zs i k M α i i )E G

48 No dependency Theorems Pauli Measurements Theorem. A unitary map is in Clifford iff a pattern implementing it with measurement angles 0 and i π 2 Theorem. If pattern P with no dependent commands implements unitary U, then U is in Clifford

49 Gottesman Knill Theorem Efficient representation in terms of Pauli Operators If the states of computation are restricted to the stabiliser states and the operation over them to the Clifford group then the corresponding quantum computation can be efficiently simulated using Classical Computing Preserves the efficient representation

50 Classical Simulation Corollary. Any MBQC pattern with only Pauli measurements can be efficiently simulated using Classical Computing. Quantum Pattern Fixing Angles Classical Pattern Model checking for a class of quantum protocols using PRISM