Universal computation by quantum walk

Transcription

1 Universal computation by quantum walk Andrew Childs C&O and IQC Waterloo arxiv:

2 Random walk A Markov process on a graph G = (V, E).

3 Random walk A Markov process on a graph G = (V, E). In discrete time: V V Stochastic matrix W R ( k W kj =1) with W kj 0 iff (j, k) E probability of taking a step from j to k

4 Random walk A Markov process on a graph G = (V, E). In discrete time: V V Stochastic matrix W R ( k W kj =1) with W kj 0 iff (j, k) E probability of taking a step from j to k Dynamics: p t = W t p 0 p t R V t =0, 1, 2,...

5 Random walk A Markov process on a graph G = (V, E). In discrete time: V V Stochastic matrix W R ( k W kj =1) with W kj 0 iff (j, k) E probability of taking a step from j to k Dynamics: p t = W t p 0 p t R V t =0, 1, 2,... Ex: Simple random walk. W kj = { 1 deg j (j, k) E 0 (j, k) E

6 Random walk A Markov process on a graph G = (V, E).

7 Random walk A Markov process on a graph G = (V, E). In continuous time: Generator matrix M R V V ( k M kj =0) with M kj 0 iff (j, k) E probability per unit time of taking a step from j to k

8 Random walk A Markov process on a graph G = (V, E). In continuous time: Generator matrix M R V V ( k M kj =0) with M kj 0 iff (j, k) E Dynamics: probability per unit time of taking a step from j to k d dt p(t) =Mp(t) p(t) R V t R

9 Random walk A Markov process on a graph G = (V, E). In continuous time: Generator matrix M R V V ( k M kj =0) with M kj 0 iff (j, k) E probability per unit time of taking a step from j to k d Dynamics: dt p(t) =Mp(t) p(t) R V t R deg j j = k Ex: Laplacian walk. M kj = L kj = 1 (j, k) E 0 (j, k) E

10 Quantum walk Quantum analog of a random walk on a graph G = (V, E). Idea: Replace probabilities by quantum amplitudes.

11 Quantum walk Quantum analog of a random walk on a graph G = (V, E). Idea: Replace probabilities by quantum amplitudes. d dt p(t) =Mp(t) p(t) R V v V p v (t) = 1

12 Quantum walk Quantum analog of a random walk on a graph G = (V, E). Idea: Replace probabilities by quantum amplitudes. d dt p(t) =Mp(t) p(t) R V v V p v (t) = 1 i d dt q(t) =Hq(t) q(t) C V v V q v (t) 2 =1

13 Quantum walk Quantum analog of a random walk on a graph G = (V, E). Idea: Replace probabilities by quantum amplitudes. d dt p(t) =Mp(t) p(t) R V v V p v (t) = 1 i d dt q(t) =Hq(t) q(t) C V v V q v (t) 2 =1 H = H with H kj 0 iff (j, k) E

14 Quantum walk Quantum analog of a random walk on a graph G = (V, E). Idea: Replace probabilities by quantum amplitudes. d dt p(t) =Mp(t) p(t) R V v V p v (t) = 1 i d dt q(t) =Hq(t) q(t) C V v V q v (t) 2 =1 H = H with H kj 0 iff (j, k) E Ex: Adjacency matrix. H kj = A kj = { 1 (j, k) E 0 (j, k) E

15 Aside: Discrete-time quantum walk We can also define a quantum walk that proceeds by discrete steps. [Watrous 99]

16 Aside: Discrete-time quantum walk We can also define a quantum walk that proceeds by discrete steps. [Watrous 99] Unitary operator U with U kj 0 iff (j, k) E

17 Aside: Discrete-time quantum walk We can also define a quantum walk that proceeds by discrete steps. [Watrous 99] Unitary operator U with U kj 0 iff (j, k) E [Meyer 96], [Severini 03]

18 Aside: Discrete-time quantum walk We can also define a quantum walk that proceeds by discrete steps. [Watrous 99] Unitary operator U with U kj 0 iff (j, k) E [Meyer 96], [Severini 03] We must enlarge the state space: C V C V instead of C V. Unitary operator U with U (k,j),(j,l) 0 iff (j, k) E

19 Aside: Discrete-time quantum walk We can also define a quantum walk that proceeds by discrete steps. [Watrous 99] Unitary operator U with U kj 0 iff (j, k) E [Meyer 96], [Severini 03] We must enlarge the state space: C V C V instead of C V. Unitary operator U with U (k,j),(j,l) 0 iff (j, k) E In this talk we will focus on the continuous-time model.

20 Quantum walk algorithms Exponential speedup for black box graph traversal [CCDFGS 03] Search on graphs [Shenvi, Kempe, Whaley 02], [CG 03, 04], [Ambainis, Kempe, Rivosh 04] Element distinctness [Ambainis 03] Triangle finding [Magniez, Santha, Szegedy 03] Checking matrix multiplication [Buhrman, Špalek 04] Testing group commutativity [Magniez, Nayak 05] Formula evaluation [Farhi, Goldstone, Gutmann 07], [ACRSZ 07], [Cleve, Gavinsky, Yeung 07], [Reichardt, Špalek 08] Unstructured search (many applications) [Grover 96],...

21 The question How powerful is quantum walk? In particular: Can it do universal quantum computation?

22 The question How powerful is quantum walk? In particular: Can it do universal quantum computation? Loosely interpreted (any fixed Hamiltonian): Yes! [Feynman 85]

23 The question How powerful is quantum walk? In particular: Can it do universal quantum computation? Loosely interpreted (any fixed Hamiltonian): Yes! [Feynman 85] But what if we take the narrowest possible interpretation? Continuous-time quantum walk on a constant-degree graph, Hamiltonian given by the adjacency matrix (no edge weights)

24 The plan Scattering theory on graphs Gate widgets Simplifying the initial state: Momentum filtering and separation

25 Scattering theory [Liboff, Introductory Quantum Mechanics]

26 Momentum states Consider an infinite line:

27 Momentum states Consider an infinite line: Hilbert space: span{ x : x Z}

28 Momentum states Consider an infinite line: Hilbert space: span{ x : x Z} Eigenstates of the adjacency matrix: x k := e ikx k [ π, π) k with

29 Momentum states Consider an infinite line: Hilbert space: span{ x : x Z} Eigenstates of the adjacency matrix: x k := e ikx k [ π, π) k with We have x A k

30 Momentum states Consider an infinite line: Hilbert space: span{ x : x Z} Eigenstates of the adjacency matrix: x k := e ikx k [ π, π) k with We have x A k = x 1 k + x +1 k

31 Momentum states Consider an infinite line: Hilbert space: span{ x : x Z} Eigenstates of the adjacency matrix: x k := e ikx k [ π, π) k with We have x A k = x 1 k + x +1 k = e ik(x 1) + e ik(x+1)

32 Momentum states Consider an infinite line: Hilbert space: span{ x : x Z} Eigenstates of the adjacency matrix: x k := e ikx k [ π, π) k with We have x A k = x 1 k + x +1 k = e ik(x 1) + e ik(x+1) = (2 cos k) x k

33 Momentum states Consider an infinite line: Hilbert space: span{ x : x Z} Eigenstates of the adjacency matrix: x k := e ikx k [ π, π) k with We have x A k = x 1 k + x +1 k = e ik(x 1) + e ik(x+1) = (2 cos k) x k so this is an eigenstate with eigenvalue 2 cos k.

34 Scattering on graphs Now consider adding semi-infinite lines to two vertices of an arbitrary finite graph: G

35 Scattering on graphs Now consider adding semi-infinite lines to two vertices of an arbitrary finite graph: G

36 Scattering on graphs Now consider adding semi-infinite lines to two vertices of an arbitrary finite graph: G Three kinds of eigenstates:

37 Scattering on graphs Now consider adding semi-infinite lines to two vertices of an arbitrary finite graph: G Three kinds of eigenstates: x, left k, sc left = e ikx + R(k)e ikx x, right k, sc left = T (k)e ikx x, left k, sc right = T (k)e ikx x, right k, sc right = e ikx + R(k)e ikx x, left κ, bd ± =(±e κ ) x x, right κ, bd ± = B ± (κ)(±e κ ) x

38 Scattering on graphs Now consider adding semi-infinite lines to two vertices of an arbitrary finite graph: G Three kinds of eigenstates: x, left k, sc left = e ikx + R(k)e ikx x, right k, sc left = T (k)e ikx x, left k, sc right = T (k)e ikx x, right k, sc right = e ikx + R(k)e ikx x, left κ, bd ± =(±e κ ) x x, right κ, bd ± = B ± (κ)(±e κ ) x

39 Scattering on graphs Now consider adding semi-infinite lines to two vertices of an arbitrary finite graph: G Three kinds of eigenstates: x, left k, sc left = e ikx + R(k)e ikx x, right k, sc left = T (k)e ikx x, left k, sc right = T (k)e ikx x, right k, sc right = e ikx + R(k)e ikx x, left κ, bd ± =(±e κ ) x x, right κ, bd ± = B ± (κ)(±e κ ) x

40 Scattering on graphs Now consider adding semi-infinite lines to two vertices of an arbitrary finite graph: G Three kinds of eigenstates: x, left k, sc left = e ikx + R(k)e ikx x, right k, sc left = T (k)e ikx x, left k, sc right = T (k)e ikx x, right k, sc right = e ikx + R(k)e ikx x, left κ, bd ± =(±e κ ) x x, right κ, bd ± = B ± (κ)(±e κ ) x It can be shown that these states form a complete, orthonormal basis of the Hilbert space, where k [ π, 0] and > 0 takes certain discrete values.

41 Scattering on graphs This generalizes to any number of semi-infinite lines attached to any finite graph. line j = line j = line j = G

42 Scattering on graphs This generalizes to any number of semi-infinite lines attached to any finite graph. line j = 1 Incoming scattering states: x, j k, sc j = e ikx + R j (k) e ikx x, j k, sc j = T j,j (k) e ikx j j line j = line j = G

43 Scattering on graphs This generalizes to any number of semi-infinite lines attached to any finite graph. line j = 1 Incoming scattering states: x, j k, sc j = e ikx + R j (k) e ikx x, j k, sc j = T j,j (k) e ikx j j line j = line j = G Bound states: x, j κ, bd ± = B ± j (κ)(±e κ ) x

44 Dynamics of scattering Solution of the quantum walk equation: i d dt ψ(t) = H ψ(t) = ψ(t) = e iht ψ(0)

45 Dynamics of scattering Solution of the quantum walk equation: i d dt ψ(t) = H ψ(t) = ψ(t) = e iht ψ(0)

46 Dynamics of scattering Solution of the quantum walk equation: i d dt ψ(t) = H ψ(t) = ψ(t) = e iht ψ(0) N y, j e iht x, j = j=1 0 π e 2it cos k y, j k, sc j k, sc j x, j d k + κ,± e 2it cosh κ y, j κ, bd ± κ, bd ± x, j

47 Dynamics of scattering Solution of the quantum walk equation: i d dt ψ(t) = H ψ(t) = ψ(t) = e iht ψ(0) N y, j e iht x, j = j=1 0 π e 2it cos k y, j k, sc j k, sc j x, j d k + κ,± e 2it cosh κ y, j κ, bd ± κ, bd ± x, j = 0 π e 2it cos k( T j,j (k)e ik(x+y) + T j,j(k)e ik(x+y)) d k + e 2it cosh κ B ± j (κ)b ± j (κ) (±e κ ) x+y κ,±

48 The method of stationary phase

49 The method of stationary phase Suppose Á(k), a(k) are smooth, real-valued functions. Then for large x, the integral e ixφ(k) a(k)dk is dominated by those values of k for which. d dk φ(k) = 0

50 The method of stationary phase Suppose Á(k), a(k) are smooth, real-valued functions. Then for large x, the integral e ixφ(k) a(k)dk d is dominated by those values of k for which. dk φ(k) = 0 In scattering on graphs, we have y, j e iht x, j 0 π e ik(x+y) 2it cos k T j,j (k) dk

51 The method of stationary phase Suppose Á(k), a(k) are smooth, real-valued functions. Then for large x, the integral e ixφ(k) a(k)dk d is dominated by those values of k for which. dk φ(k) = 0 In scattering on graphs, we have y, j e iht x, j 0 π e ik(x+y) 2it cos k T j,j (k) dk The phase is stationary for k satisfying x + y + l j,j (k) =v(k)t v(k) := d dk 2 cos k = 2 sin k group velocity l j,j (k) := d dk arg T j,j (k) effective length

52 Finite lines suffice To obtain a finite graph, truncate the semi-infinite lines at a length O(t), where t is the total evolution time. This gives nearly the same behavior since the walk on a line has a maximum propagation speed of 2. E.g., from stationary phase: v(k) =2 sin k 2.

53 Computation by scattering Encode quantum circuits into graphs.

54 Computation by scattering Encode quantum circuits into graphs. Computational basis states correspond to lines ( quantum wires ).

55 Computation by scattering Encode quantum circuits into graphs. Computational basis states correspond to lines ( quantum wires ). Ex: With two qubits, we use four wires:

56 Computation by scattering Encode quantum circuits into graphs. Computational basis states correspond to lines ( quantum wires ). Ex: With two qubits, we use four wires: Quantum information propagates from left to right.

57 Computation by scattering Encode quantum circuits into graphs. Computational basis states correspond to lines ( quantum wires ). Ex: With two qubits, we use four wires: Quantum information propagates from left to right. To perform gates, attach graphs along/connecting the wires.

58 A universal gate set Theorem. Any unitary operation on n qubits can be approximated arbitrarily closely by a product of gates from the set ( ) ( ) , , i [Boykin et al. 00] We can implement these elementary gates (and indeed, any product of these gates) by scattering on graphs.

59 Controlled-not

60 Controlled-not in 01 in 10 in 11 in 00 out 01 out 10 out 11 out

61 Phase ( ) i

62 Phase ( ) i (b) in out

63 Phase ( ) i T in,out (k) = i cos 2k csc 3 k sec k 1 (b) in out π 3π 4 π 2 π k

64 Phase ( ) i 8 T in,out (k) = 8 + i cos 2k csc 3 k sec k l in,out ( π/4) = 1 1 (b) in out π 3π 4 π 2 π k

65 A basis-changing gate

66 A basis-changing gate 0 in 0 out 1 in 1 out

67 A basis-changing gate 0 in 1 in 0 out 1 out e ik (cos k + i sin 3k) T 0in,0 out (k) = 2 cos k + i(sin 3k sin k) 1 T 0in,1 out (k) = 2 cos k + i(sin 3k sin k) R 0in (k) =T 0in,1 in (k) = e ik cos 2k 2 cos k + i(sin 3k sin k) π 3π 4 π 2 π k

68 A basis-changing gate 0 in 1 in 0 out 1 out e ik (cos k + i sin 3k) T 0in,0 out (k) = 2 cos k + i(sin 3k sin k) 1 T 0in,1 out (k) = 2 cos k + i(sin 3k sin k) R 0in (k) =T 0in,1 in (k) = e ik cos 2k 2 cos k + i(sin 3k sin k) At k = ¼/4 this implements the unitary transformation U = 1 ( ) i i from inputs to outputs π 3π 4 π 2 π k

69 A basis-changing gate 0 in 1 in 0 out 1 out e ik (cos k + i sin 3k) T 0in,0 out (k) = 2 cos k + i(sin 3k sin k) 1 T 0in,1 out (k) = 2 cos k + i(sin 3k sin k) R 0in (k) =T 0in,1 in (k) = e ik cos 2k 2 cos k + i(sin 3k sin k) ( ) ( )( ) i At k = ¼/40this i 1 i 0 i implements the unitary transformation U = 1 ( ) i i = e iφ 1 ( ) from inputs to outputs π 3π 4 π 2 π k

70 Tensor product structure To embed an m-qubit gate in an n-qubit system, simply include the gate widget 2 n m times, once for every possible computational basis state of the n m qubits not acted on by the gate.

71 Tensor product structure To embed an m-qubit gate in an n-qubit system, simply include the gate widget 2 n m times, once for every possible computational basis state of the n m qubits not acted on by the gate. Note: The graph has 2 n poly(n) vertices (exponentially many), corresponding to the dimension of the Hilbert space used by the simulation. Vertices correspond to basis states, not qubits. Despite its exponential size, the graph has a succinct description in terms of the circuit being simulated. In particular, the quantum walk can be efficiently simulated by a universal quantum computer.

72 Composition law To perform a sequence of gates, simply connect the output wires to the next set of input wires.

73 Composition law To perform a sequence of gates, simply connect the output wires to the next set of input wires. Arrange the transmission/reflection coefficients as transformations from inputs to outputs: { R jin j = j T j,j = T jin,j out R j,j = T jin,j in j j { R jout j = j T j,j = T jout,j R in j,j = T jout,j out j j

74 Composition law To perform a sequence of gates, simply connect the output wires to the next set of input wires. Arrange the transmission/reflection coefficients as transformations from inputs to outputs: { R jin j = j T j,j = T jin,j out R j,j = T jin,j in j j { R jout j = j T j,j = T jout,j R in j,j = T jout,j out j j Then we have T 12 = T 1 (1 R 2 R1 ) 1 T 2 R 12 = R 1 + T 1 (1 R 2 R1 ) 1 R 2 T1 T 12 = T 2 (1 R 1 R 2 ) 1 T1 R 12 = R 2 + T 2 (1 R 1 R 2 ) 1 R1 T 2

75 Example H 00 in 00 out 01 in 10 in 01 out 10 out 11 in 11 out

76 Example in action

77 Simplifying the initial state So far, we have assumed that the computation takes place using only momenta near k = ¼/4.

78 Simplifying the initial state So far, we have assumed that the computation takes place using only momenta near k = ¼/4. Can we relax this restriction? Start from a single vertex of the graph?

79 Simplifying the initial state So far, we have assumed that the computation takes place using only momenta near k = ¼/4. Can we relax this restriction? Start from a single vertex of the graph? Idea: A single vertex has equal amplitudes for all momenta. Filter out momenta except within 1/poly(n) of k = ¼/4.

80 Momentum filter (d) in out

81 Momentum filter 1 (d) in out 1 4 π 3π 4 π 2 π k

82 Transfer matrix To create a narrow filter, repeat the basic filter many times in series.

83 Transfer matrix To create a narrow filter, repeat the basic filter many times in series. This can be analyzed using a transfer matrix approach. ( ) ( ) x +1 k, sc in x k, sc x k, sc in = M in Write x 1 k, sc in

84 Transfer matrix To create a narrow filter, repeat the basic filter many times in series. This can be analyzed using a transfer matrix approach. ( ) ( ) x +1 k, sc in x k, sc x k, sc in = M in Write x 1 k, sc in For m filters, suppose ( ) M m a b = c d Then T = 2ie ikm sin k ae ik b + c + de ik

85 Transfer matrix To create a narrow filter, repeat the basic filter many times in series. This can be analyzed using a transfer matrix approach. ( ) ( ) x +1 k, sc in x k, sc x k, sc in = M in Write x 1 k, sc in For m filters, suppose ( ) M m a b = c d Then T = 2ie ikm sin k ae ik b + c + de ik Eigenvalues of M π 3π 4 π 2 π k

86 The curse of symmetry Problem: Our filter passes k = 3¼/4 in addition to k = ¼/4.

87 The curse of symmetry Problem: Our filter passes k = 3¼/4 in addition to k = ¼/4. Generically, distinct momenta propagate at different speeds; but v( π/4) = 2 sin(π/4) = 2 v( 3π/4) = 2 sin(3π/4) = 2

88 The curse of symmetry Problem: Our filter passes k = 3¼/4 in addition to k = ¼/4. Generically, distinct momenta propagate at different speeds; but v( π/4) = 2 sin(π/4) = 2 v( 3π/4) = 2 sin(3π/4) = 00!in2! 00out (b) 00in 00 (a) (a) (b) out! ny stationary and the of the firstisterm is points, andpoints, the phase of phase the first term 01!in! 01in! (k) 2t cos k, which is staven by k(x + y) + arg T j,j y) + arg Tj,j! (k) 2t cos k, which is sta 10!in! 10in onary In for fact, all widgets so far have a symmetry3 under 11! x + y +!j,j! (k) = v(k)t,! (k) x + y +(a)!j,j 00 00out! (b) in! = v(k)t, 00in! 00out! (a) 01 (b)! 01! here out in 01! 01out! 10inin! 10out! in! 10in! d 10out! in! 11in:=! 11 2 cos kout =! 2 sin k v(k) 11in! dk 11out! d 2 cos k = 2 sin k (k) := the 0 group atout momentum! (c) (d) k, and in! dkvelocity 0 (c) 0in! 0out! (d) d and ocity k, 1inat! momentum 1! out!! (k) := arg T 1in! j,j 1out dk! j,j! (k) (8) (8) out! out! (9) (9) 3 (c) 0in! (c) 0in! 1in! 1in! 11ink!in 01out! out! 01 10out!! in! 10 in! out 11 out! k. π 11out! 0out! 0out! (d) (e) (d) (e) 1out! 1out! out! out! in! in! out! in! in! out! FIG. 1: Widgets used to construct a universal quantum (e) (e) puter. Open circles indicate vertices where previous o FIG. cessive 1: Widgets toattached. construct universal quant widgetsused can be (a)acontrolled-note gat in! (10) out! Phase shift. (c) Basis-changing gate. where (d) Momentum puter. Open circles indicate vertices previou in! out! (e)widgets Momentum cessive canseparator. be attached. (a) Controlled-note d in! through out! the path G from j!j,j!effective (k) := length argoftthe (10) line Phase j,j! (k) in! out! shift. (c) Basis-changing gate. (d) Momentu dkfor large x + y we have [25, Eq. 3.2] line j!.4 Then (e) Momentum separator. IG. 1: Widgets used to construct a universal quantum comwith two single-qubit gates that generate a dense s! length of the path through G from line j FIG. 1: Widgets used to construct universal quantum com Tj,j! (ka )! iht

89 The curse of symmetry Problem: Our filter passes k = 3¼/4 in addition to k = ¼/4. Generically, distinct momenta propagate at different speeds; but v( π/4) = 2 sin(π/4) = 2 v( 3π/4) = 2 sin(3π/4) = 00!in2! 00out (b) 00in 00 (a) (a) (b) out! ny stationary and the of the firstisterm is points, andpoints, the phase of phase the first term 01!in! 01in! (k) 2t cos k, which is staven by k(x + y) + arg T j,j y) + arg Tj,j! (k) 2t cos k, which is sta 10!in! 10in onary In for fact, all widgets so far have a symmetry3 under 11! x + y +!j,j! (k) = v(k)t,! (k) x + y +(a)!j,j 00 00out! (b) in! = v(k)t, 00in! 00out! (a) 01 (b)! 01! here out in 01! 01out! 10inin! 10out! in! 10in! d 10out! in! 11in:=! 11 2 cos kout =! 2 sin k v(k) 11in! dk 11out! (8) (8) out! out! (9) 3 (c) 0in! (c) 0in! d 2 cos k = 2 sin k (9) (k) := the 0 group atout momentum andall bipartite.! dkvelocity 0! (c) (d) k, (e) in This is because they are (c) 0in! 0out! (d) d and ocity k, 1inat! momentum 1! out!! (k) := arg T 1in! j,j 1out dk! j,j! (k) 1in! 1in! 11ink!in 01out! out! 01 10out!! in! 10 in! out 11 out! k. π 11out! 0out! 0out! (d) (e) (d) (e) 1out! 1out! out! out! in! in! out! in! in! out! FIG. 1: Widgets used to construct a universal quantum (e) puter. Open circles indicate vertices where previous o FIG. cessive 1: Widgets toattached. construct universal quant widgetsused can be (a)acontrolled-note gat in! (10) out! Phase shift. (c) Basis-changing gate. where (d) Momentum puter. Open circles indicate vertices previou in! out! (e)widgets Momentum cessive canseparator. be attached. (a) Controlled-note d in! through out! the path G from j!j,j!effective (k) := length argoftthe (10) line Phase j,j! (k) in! out! shift. (c) Basis-changing gate. (d) Momentu dkfor large x + y we have [25, Eq. 3.2] line j!.4 Then (e) Momentum separator. IG. 1: Widgets used to construct a universal quantum comwith two single-qubit gates that generate a dense s! length of the path through G from line j FIG. 1: Widgets used to construct universal quantum com Tj,j! (ka )! iht

90 Momentum separator (e) in out

91 Momentum separator T in,out (k) = [ 1+ ] 1 i(cos k + cos 3k) sin k + 2 sin 2k + sin 3k sin 5k (e) 1 in out π 3π 4 π 2 π k

92 Momentum separator T in,out (k) = [ 1+ ] 1 i(cos k + cos 3k) sin k + 2 sin 2k + sin 3k sin 5k l in,out ( π/4) = 4(3 2 2) l in,out ( 3π/4) = 4( ) 23.3 (e) 1 in out π 3π 4 π 2 π k

93 A universal computer Consider an m-gate quantum circuit (unitary transformation U).

94 A universal computer Consider an m-gate quantum circuit (unitary transformation U). Graph: log Θ(m 2 ) filter widgets on input line Momentum separation widget on input line Widgets for m gates in the circuit Truncate input wires to length Θ(m 4 )

95 A universal computer Consider an m-gate quantum circuit (unitary transformation U). Graph: log Θ(m 2 ) filter widgets on input line Momentum separation widget on input line Widgets for m gates in the circuit Truncate input wires to length Θ(m 4 ) Simulation: Start at vertex x = Θ(m 4 ) on input line t = π (x + l)/ 2π = O(m 4 ) Evolve for time Measure in the vertex basis Conditioned on reaching vertex 0 on some output line s (which happens with probability Ω(1/m 4 )), the distribution over s is approximately s U

96 Applications? Quantum algorithms Quantum complexity theory Architectures for quantum computers