A Glimpse of Quantum Computation Zhengfeng Ji (UTS:QSI) QCSS 2018, UTS 1. 1
Introduction What is quantum computation? Where does the power come from? Superposition Incompatible states can coexist Transformation of superpositions Fourier transform Quantum walk Hamiltonian simulation Intuition 2. 1
Superposition Lecture by Allan Adams https://www.youtube.com/watch?v=lz3bpuko5zc Superposition of function calls f : {0,1} n {0,1} m Assume that has a classical circuit of size polynomial in n, then the following transformation is ef cient on a quantum computer 1 2 n x x,0 1 2 n x x, f(x) 2. 2
Quantum Fourier Transform 3. 1
Let Discrete Fourier Transform (DFT) X = {0,1,,N 1} be a nite set The discrete Fourier transform transforms a function f^ f : X C to another function where f^ 1 (y) = ω xy f(x). N N x X A transformation on a sequence of N complex numbers Important tool in classical signal processing 3. 2
Fast Fourier Transform (FFT) Cooley Tukey algorithm, from f^ 1 (y) = N x X Gauss 1805! Assume for simplicity N = 2 n O( ) Recursive structure: for, N 2 to O(N log N) ω xy N 0 y < N/2 f(x). [ (y) = f^ (y)] e(y) + ω y N f^ o f^ 1 2 3. 3
What about the second half? f^ 0y ( ) 1 [ (y)] = (y) + 2 f^ e ω y N f^ o f^ 1y ( ) 1 [ (y)] = (y) 2 f^ e ω y N f^ o FFT: compute the fast Fourier transform (recursively) on the even and odd parts respectively and combine the results using the above rule 3. 4
Quantum Fourier Transform (QFT) For function f : X C, de ne quantum state f = f(x) x. x X Quantum Fourier transform maps the superposition f to f^ Recall that quantum computation is about the processing of superposition states In QFT, we transform the amplitudes by DFT The QFT is a unitary transform F = 1 N x,y X ω xy N y x 3. 5
QFT: E cient Realization Use the recursion in a quantum way f^ 0y ( ) f^ 1y ( ) 1 [ (y)] = (y) + 2 f^ e ω y N f^ o 1 [ (y)] = (y) 2 f^ e ω y N f^ o Apply QFT of qubits to the rst qubits only once!!! n 1 n 1 Then do some post-processing Cheating?! No! 3. 6
1. After the QFT on the rst qubits, we have 2. De ne function Apply n 1 f = 0 + 1 0 + 1 fe fo fe ^ fo ^ go(y) ^ = ω y (y) N f^ o n 1 controlled phase gates, we have fe ^ 0 + 1 go ^ 3. Apply Hadamard to the last qubit, we get + fe ^ go ^ 0 fe + ^ go ^ 1 2 2 4. Move the last qubit to the rst by SWAP gates and we have f^
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Remarks on QFT We have shown an ef cient implementation of QFT Complexity or Discard small rotations, gives an approximate QFT Complexity n 2 nlog n log 2 N U i V i ϵ Lemma. For unitary operators and,, we have U t U2U1 V t V2V1 tϵ. U i V i Do NOT expect it to be useful in most problems of signal processing You cannot read out the values of the amplitudes! 3. 8
Phase Estimation In phase estimation, we are given access to controlled-u x for all x X and also an eigenstate ϕ of U U ϕ iϕ = e ϕ The problem is to estimate the phase ϕ some desired precision [0, 2π) to Inverse Fourier transform F 1 1 = N xy ω N y x x,y X 3.9
n ϕ To get an -bit estimate of, the phase estimation procedure 1 x ϕ 1. Prepare the state N x X 2. Apply unitary operator x x x X U x The state becomes 1 N x X e iϕx x ϕ 3. Apply an inverse quantum Fourier transform on the rst register ϕ = 2πy/2 n y X If for some, the state in the rst register before the inverse Fourier transform is F y The rst register is a good approximation of ϕ/2π 3. 10
An Example: Discrete Log Given a cyclic group g0 generated by g0 and an element, the size is known h G The discrete logarithm of in is the smallest non- such that with respect to g0 negative integer G = N = G α log g0 h h G g α 0 = h Important for classical cryptography (Dif e- Hellman key exchange) No ef cient classical algorithm is known 3. 11
Quantum Algorithm for Discrete Log Consider the following unitary U : g hg U x It is possible to implement controlled- operators for any x by repeated squaring De ne states 1 = ϕ y ω xy, then N N gx 0 x X 1 ϕ y ω xy N g x+ log h g 0 0 ω y log h g 0 N N x X Use phase estimation! We can sample a random pair of (y, ϕ ) y Factoring is in BQP U = = ϕ y
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Quantum Walk 4. 1
Let Continuous-time Random Walk on Graph G = (V, E) be a graph The Laplacian of is a matrix of size where The continuous-time random walk on the differential equation The distribution evolves as L G V V L j,k = deg(j) 1 0 d dt p j (t) = p(t) = e Lt G k V j = k (j, k) E (j, k) E is de ned as the solution of p(0) L jk p k (t) 4. 2
Schrödinger equation: Continuous-time Quantum Walk Continuous-time quantum walk is the quantum time evolution de ned by the Laplacian From random transformations of probability distributions to transformations of superpositions Probability to amplitudes: For i d dt ψ(t) = Hψ(t) q j (t) = j ψ(t) d i (t) = (t) dt q j k V L jk q k, we have Continuous-time random and quantum walks exhibit very different properties! 4. 3
Discrete-time Random Walk In a discrete-time random walk on graph to its neighbors with equal probability G, we move from a vertex The walk is de ned by the matrix M jk ={ 1/deg(k) 0 If the probability distribution is after one step of random walk is (j, k) E otherwise p p, the distribution = Mp 4. 4
Discrete-time Quantum Walk It is not easy to come up with a proper de nition j? 1 = k j is not a unitary map! deg(j) k:(j,k) E The key idea to solves this problem is to enlarge the Hilbert space Walk on the edges, instead of the vertices (Watrous) j C = j j (2 I) j j j V j U = SC S Leaves invariant The quantum walk operator where swaps the two vertex registers 4. 5
Szegedy Walk: Markov Chain Quantized In general, for any stochastic matrix whose entry is the probability of moving from to De ne state De ne operator j=1 Quantum walk operator for one step is j N k P = j k ψ j P kj k=1 N Π = ψ j ψ j P kj U = S(2Π I) 4. 6
Spectrum Theorem Theorem. For any stochastic matrix, de ne matrix and let be the complete set of D jk = P jk P kj eigenvectors of ±i and arccos λ. D { λ } P. Then the discrete-time quantum walk corresponding to have eigenvalues U = S(2Π I) P ±1 e 4. 7
Recall that we have a Markov chain De ne isometry Check that For state k V and we de ned states λ ~, check that U = S, and US = 2λS λ ~ λ ~ λ ~ λ ~ λ ~ The two-dimensional subspace spanned by and is invariant under. Coincidence? U P = j, k ψ j P kj T = ψ j = j, k j j P kj j V j,k V T T T T = I, T = Π, ST = D = T λ λ ~ S λ ~
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Hitting Time G G M Given graph, decide if has a marked vertex in or not Consider classical Markov chain simplicity that is symmetric P P P and assume for De ne as but stops if one reaches a marked vertex P =( 0) =( ), ( P ) t P t M P M Q I Starting from a uniform distribution on not nding a marked vertex after steps is P t 0 Q I V M 1, = N M P t 1 M M PM t P M, the probability of t 4. 9
Probability of not nding a marked vertex is at most P M t A lemma that bounds the norm of P M Lemma. If the second largest eigenvalue of is at most, and, then. M ϵn 1 δϵ P M P 1 δ To bring P M t = O(1/δϵ) t to some constant below some constant, we need, so the classical hitting time is O(1/δϵ) 4. 10
Quantum Hitting Time Consider the quantum walk operator corresponding to, and State ψ 1 = ψ j N M j M The matrix for the Markov chain is U P D P ( P M0 0) U ψ Phase estimation of on state solves the decision problem If M is empty, we have P = P ψ and is a 1-eigenstate of U, and the phase estimation always outputs 0 ψ Otherwise, when M is not empty, the state lives in the ±i subspace with eigenvalues e arccos λ Time complexity: 1/arccos λ = O(1/ δϵ) steps of quantum walk I 4. 11
Quantum Search In unstructured search, one is given black-box access to a function f : S {0,1} where S is a nite set of N elements The set is the set of marked items Consider the random walk on the complete graph of vertices Spectral gap of M = {x S : f(x) = 1} Quantum hitting time P is δ = N/(N 1) Slightly different from Grover's search algorithm P N P = S S I N 1 1 N 1 O(1/ δϵ) = O( N/ M ) N 4. 12
Quantum Hamiltonian Simulation 5. 1
Schrödinger equation Hamiltonian Simulation i d dt ψ(t) = H ψ(t) H 2 n 2 n, mathematically a Hermitian matrix of size by, is called the Hamiltonian of the system and, as the equation says, it governs the dynamics of system ψ(t) = ψ(0) U t = We have for unitary operator Give Hamiltonian H and time t, nd a quantum circuit U consisting of gates such that U t poly(n, t,1/ϵ) U e iht ϵ e iht 5. 2
Product Formula If both H1 and H2 are ef ciently simulated, then so is H1 + H2 Notice that e A+B may be different from e A e B as A and B may not commute with each other Lie product formula m = O((νt e A+B = lim ( N ea/n e B/N ) N ) 2 /ϵ) ν = max(, ) For, where H1 H2, we have t/m i t/m i( + )t ( H 1 e H 1 ) m e ϵ e i H 1 H2 5. 3
Local Hamiltonians can be ef ciently simulated H = j H j term qubits H H j where each acts on at most k can be speci ed ef ciently 5. 4
An only N N Sparse Hamiltonian Simulation Hermitian matrix is sparse if in any xed row there are nonzero entries d = poly(log N) Assume there is a classical algorithm that given the row index, outputs the list of indices and values of the nonzero entries Simple case: H is diagonal (Not a small sum) x,0 x, H x,x x, e ith x,x H x,x x,0 e ith x,x = ( x ) 0 e iht 5. 5
Graph Edge Coloring Let G be an undirected graph of maximum degree An edge-coloring of at most colors can be ef ciently computed for bipartite Simulate the Hamiltonian for different color respectively and use Lie product formula y = Let be the vertex adjacent to by an edge of color De ne max x v c (x) c min = min(x, y) x = max(x, y) d 2 and V 2 2 x min, max x x H x,c Let be the by unitary acting on the space spanned by de ned by the Hamiltonian G x c d 5. 6
De ne two quantum circuits : x,0,,, x U1 min max x x V where x is a description of by unitary 2 V x V acts on the span of x and Consider U 1 and assume V x 2 :,, ϕ,, ϕ U2 min max x x V min max x x x V x V x min max x x where x 1 U2U1 = α + β V min max x x x x,0 min, max,, x x x V x min, max, x x x V x V x min max x x V min min max x x x x V max x x min max x x = α,,, + β,,, α + β By linearity U1 1 U2U1 simulate the color c part of the Hamiltonian
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More E cient Methods All known product formula based algorithms have complexity superlinear in t, is it possible to achieve linear complexity? Better performance in Connections to continuous-time and discrete-time quantum walks De ne isometry polylog 1 ϵ N j = j, k + H j, N + jk 1 ψ j X k=1 H 1 1 N jk X k=1 A unitary operator motived by the quantum walk operator that has ±i eigenvalues arccos λ e 5. 8
Phase estimation ψ 1. Apply the isometry T to state 2. Perform the phase estimation on the quantum walk operator with some precision δ 3. Compute an estimate of λ from the estimated value of 4. Introduce the phase e iλt 5. Uncompute everything Quantum signal processing Reading material: HHL algorithm Hamiltonian simulation + Phase estimation arccos λ [Low, Chuang, PRL 118 010501] [Harrow, Hassidim, Lloyd, PRL 103 150502] 5. 9
Summary A glimpse of quantum computation through Quantum Fourier transform f f^ Discrete-time quantum walk = S(2Π I) ψ t 1 ψ t ψ t 1 Hamiltonian simulation ψ(0) ψ(t) = ψ(0) e iht 6. 1