Quantum Fan-out is Powerful Robert ˇ Spalek, CWI (joint work with Peter Høyer, Calgary) 1
Quantum circuits resemble classical reversible circuits: � Number of (qu)bits stays constant during the H computation. H � Reversible gates are ordered into layers and X H applied in the corresponding order. time Differences: � State of computation is a unit vector instead of value 0 , 1 ,..., 2 n − 1 . � Gate is a unitary mapping on some subspace instead of a permutation of the values. 2
Quantum fan-out Motivation: small decoherence time . � We want to minimise the depth of the circuit: 1. Gates on different qubits can be applied in parallel. 2. Commuting gates can be applied on the same qubits in parallel. � We allow unbounded quantum fan-out gate: • It behaves like a controlled-not-not-. . . -not gate: | x �| y 1 � ... | y n � → | x �| y 1 ⊕ x � ... | y n ⊕ x � . • This is not quantum cloning! 3
Physical implementation � Interaction between more than two qubits in principle possible in ion-trap and NMR models. � [Fenner, 2003] Fan-out implemented by a Hamiltonian with number of terms quadratic in n . 4
[Moore, 1999] Parity in constant depth Parity and fan-out can simulate each other. H H H H H H H H = = = H H H H H H H H 2 � Hadamard gates change the direction of controlled-not. � � � Two applications of H = 1 1 1 cancel, i.e. H 2 = I . √ − 1 1 2 5
Parameters of the circuit model We investigate operators computed by uniform families of circuits: � depth bounded by d ( n ) , mostly constant, � polynomial size, � fixed basis of one-qubit gates: • Hadamard gate H , • R z ( ϕ ) for ϕ irrational multiple of π , and unbounded fan-out gate, � described by a log-space Turing machine. 6
Parallelisation method Gates can be applied on the same qubits in parallel whenever: 1. they commute , and 2. we know the basis in which they all are diagonal (there is always such a basis), and 3. we can efficiently change into this basis. Advantages: Disadvantages: smaller depth needs ancilla qubits gates can be controlled needs basis change 7
1. Changing the basis n . . . k T † T † T † T † T T T T U 1 U 2 U n . . . V 1 V 2 V n � Put TT † = I between U k and U k + 1 . � Take V k = T † U k T as new operators. They are diagonal in the computational basis. 8
2. Parallelising diagonal operators n . . . V 1 T † k T | 0 � | 0 � V 2 n . . . | 0 � | 0 � V n � Fan-out creates/destroys n entangled copies of target qubits. � V k are diagonal, so they just impose phase shifts. � These phase shifts multiply and thus can be applied in parallel. 9
[Moore, 1999] mod[k] in constant depth � The number | x | mod k can be computed in this way: • Initialise ancilla counter y to 0 , this is ⌈ log k ⌉ qubits. • Each input bit x k controls one increment of y modulo k . • At the end: y = | x | mod k . � The increment gates commute, so can be parallelised. k is fixed , hence the basis change and the increments can be computed exactly in constant depth. 10
Rotation by Hamming weight | x 0 � | x 0 � | x 1 � | x 1 � . . . | x n − 1 � | x n − 1 � Define: � 1 | µ | x | ϕ � R z ( ϕ ) | 0 � H H � 0 R z ( ϕ ) : = e i ϕ 0 R z ( ϕ ) 1 + e i ϕ w | 0 � + 1 − e i ϕ w | 0 � | µ w ϕ � : = | 1 � 2 2 ancillas . . . R z ( ϕ ) | 0 � | 0 � + e i ϕ | x | | 1 � � � = | µ | x | √ The circuit maps | 0 � to H ϕ � 2 in depth 5 and linear size. 11
Approximate circuit for Or | y � | z � | µ | x | | µ | y | m 0 � m � | x 1 ... x n � 2 π 2 π | µ | x | m 1 � | 00 ... 0 � 2 π . . . | µ | x | m ( m − 1 ) � | 00 ... 0 � 2 π After the first set of rotations, either | y | = 0 or | y | ≈ m 2 . � � n 2 log n The circuit has constant depth and size O ( mn ) = O . 12
1st layer of the circuit for Or � Let m = n log n . For all k ∈ { 0 , 1 , 2 ,..., m − 1 } , ϕ k � for angle ϕ k = 2 π compute in parallel | y k � = | µ | x | m · k . � If | y k � is measured, the expected value is 2 1 − e i ϕ k | x | � � = 1 − cos ( ϕ k | x | ) � � E [ Y k ] = � � � 2 � 2 � � and the expected Hamming weight of | y � = | y m − 1 ... y 1 y 0 � is m − 1 � 2 π k � � 0 if | x | = 0 , E [ | Y | ] = m 2 − 1 ∑ m | x | = cos m if | x | � = 0 . 2 2 k = 0 � ≥ ε m � | Y |− m 1 �� � � � Moreover, if | x | � = 0 , then P ≤ 2 ε 2 m . 2 13
2nd layer of the circuit for Or � The register | y � is not directly measured, but its Hamming weight controls another rotation on a new ancilla qubit | z � . � Compute | z � = | µ | y | 2 π / m � . Let Z be the outcome after | z � is measured. • If | x | = 0 , then | y | = 0 and Z = 0 with certainty. � > m � | y |− m 1 2 log n = 1 1 � � • It | x | � = 0 , then √ n with probability < 2 m / n = n . 2 � ≤ m � | y |− m � � • If √ n , then Z = 1 with high probability and 2 1 − cos 2 π 1 + e i 2 π 2 � � √ n m | y | � � 1 � � P [ Z = 0 ] = ≤ = O . � � 2 2 n � � � if | x | = 0 , 1 � Hence P [ Z = 0 ] = � � 1 if | x | � = 0 . O n 14
Remarks on the Or gate � The error is bounded by 1 n and one-sided. 1 If we need small error n c , we create c copies and compute the exact Or of them. This can be done in log c = O ( 1 ) layers. � � π k � The construction uses rotations R z for arbitrary k , m . m We are only allowed to use a fixed set of one-qubits gates. • Every rotation can be approximated with polynomially small � √ � 2 π · q error by R z for a polynomially large q . • q iterations can be done in parallel, so depth is preserved. 15
Generalisation: exact[q] gate � Or gate tests whether | x | = 0 . exact[q] gate tests whether | x | = q . � | x 0 � | x 0 � | x 1 � | x 1 � • Can be computed similarly to Or. . . . | x n − 1 � | x n − 1 � • Add rotation R z ( − ϕ q ) to the first layer | µ | x |− q R z ( ϕ ) � | 0 � ϕ H H and obtain | µ | x |− q � instead of | µ | x | ϕ � . ϕ R z ( ϕ ) | 0 � • The second layer stays the same. . . . • Measure output qubit | z � and get R z ( ϕ ) | 0 � added � if | x | = q , 1 rotation R z ( – ϕ q ) | 0 � P [ Z = 0 ] = � � 1 if | x | � = q . O n � exact[q] gates can be used for threshold[t] and counting gates. 16
Arithmetics and sorting in constant depth [Siu et al., 1993] The following functions are computed by constant depth threshold circuits: 1. summation and multiplication of n integers, 2. division of two integers, 3. and sorting of n numbers. The construction uses weighted threshold gates . Quantum circuits with fan-out can approximate also the weighted threshold gate in constant depth. 17
Exact computation of Or and exact[q] Exact reduction of Or on n qubits to Or on log n qubits: � Let m = ⌈ log ( n + 1 ) ⌉ . For all k ∈ { 1 , 2 ,..., m } , ϕ k � for angle ϕ k = 2 π compute in parallel | y k � = | µ | x | 2 k . • If | x | = 0 , then | y k � = | 0 � for each k . • If | x | � = 0 , decompose it into | x | = 2 a ( 2 b + 1 ) and � 1 | y a + 1 � = 1 − e i ϕ a + 1 | x | = 1 − e i π ( 2 b + 1 ) = 1 − e i π = 1 . 2 2 2 � It follows that | x | = 0 ⇐ ⇒ | y | = 0 . The reduction is exact, the depth is O ( 1 ) , and the size is O ( n log n ) . After O ( log ∗ n ) iterations, the number of qubits is constant. 18
Randomised vs quantum depth Problem Randomised Quantum Θ ( log n ) O ( log ∗ n ) Or and threshold[t] exactly Θ ( log n ) Θ ( 1 ) mod[k] exactly Θ ( loglog n ) Θ ( 1 ) Or with error 1 n Ω ( loglog n ) Θ ( 1 ) threshold[t] with error 1 n � Classical lower bounds are for the model with bounded fan-in of Or and unbounded parity . (Proven by the polynomial method and Yao’s principle.) � Quantum upper bounds are for the model with bounded fan-in and unbounded fan-out . The exact algorithm for Or uses arbitrary one-qubit gates, though. 19
Possible improvements? � � n 2 log n 1. We can reduce the size of circuit for Or from O to O ( n log n ) , O ( n loglog ... log n ) , or even O ( n log ∗ n ) ! Can it be made linear? 2. Exact circuit for Or of constant depth? 3. Exact circuit for Or of sub-logarithmic depth with a fixed basis of one-qubit gates? 20
Quantum Fourier transform (QFT) Performs Fourier transform on the amplitudes of the state: 2 n − 1 1 ∑ e 2 π ixy / 2 n | y � . F : | x � → | ψ x � = √ 2 n y = 0 � n 2 � � [Shor, 1994] Compute QFT in depth O ( n ) , size O , without ancillas. � [Cleve & Watrous, 2000] Approximate QFT with error ε � � log n + loglog 1 n log n � � in depth O and size O . ε ε � [Høyer & ˇ Spalek, 2002] Using fan-out, approximate QFT with polynom. small error in constant depth and polynomial size. 21
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