# Maximum Searching Algorithm¶

An Introduction to Maximum Searching Algorithm

The algorithm for searching for the maximum value solves the problem of giving a data set and finding the element with the largest value. On a classic computer, to complete such a task requires a traversal of the database, the complexity is $$O(N)$$. The quantum algorithm for searching for the maximum value is based on the Grover search algorithm. It was proposed by Christoph Durr and Peter Hoyer in January 1999 [A quantum algorithm for finding the minimum] (https://arxiv.org/pdf/quant-ph/9607014). Ashish Ahuja and Sanjiv Kapoor then made a better complexity analysis and yielded a bound than the original one of $$15\sqrt{N}$$ [A Quantum Algorithm for finding the Maximum] (https://arxiv.org/abs/quant-ph/9911082v1). The complexity of the algorithm is $$O(\sqrt(N)),$$ which is quadratic acceleration compared to the classical algorithm. Below we briefly introduce this quantum algorithm.

Basic idea of the quantum algorithm for searching the maximum value: For the search space A (a dataset), select one of the elements as the current maximum element randomly, and use the improved Grover search algorithm, that is, the Grover algorithm that does not know the number of target elements, to search for a larger element in A. Then replace the maximum element with the new one. The expected value of the total replacing times to get the maximum element is no more than $$O(log N)$$. After finding the maximum element, the Grover algorithm cannot find a new element. Then the algorithm stops and outputs the maximum element.

Improved Grover search algorithm (Grover algorithm without knowing the number of target elements)

As can be seen from the above description, the improved Grover search algorithm plays an important role in the algorithm for searching for the maximum. We use the HiQ demo to implement an improved Grover search algorithm.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 def run_grover(eng, Dataset, oracle, threshold): """ Args: eng (MainEngine): Main compiler engine to run Grover on. Dataset(list): The set to search for an element. oracle (function): The oracle black box, an n-qubit register. It is used to flip the relative phase for the correct bit string. threshold: The threshold, where the algorithm stops and outputs the local of the element in the database that is greater than the threshold. Returns: solution (list): the location of Solution. """ N = len(Dataset) n = math.ceil(math.log2(N)) # number of maximum iterations we may run: num_it = int(math.sqrt(N)*9/4) # two arguments m = 1 landa = 6/5 # run i iterations i = 0 while i - |1>) s.t. a bit-flip turns into a (-1)-phase. oracle_out = eng.allocate_qubit() X | oracle_out H | oracle_out # run j iterations with Loop(eng, j): # oracle adds a (-1)-phase to the solution oracle(eng, x, Dataset,threshold, oracle_out) # reflection across uniform superposition with Compute(eng): All(H) | x All(X) | x with Control(eng, x[0:-1]): Z | x[-1] Uncompute(eng) # measure All(Measure) | x Measure | oracle_out # read the measure value k = 0 xvalue = 0 while kthreshold: return xvalue m = m*landa i = i+1 

For the function Oracle, the code for its implementation is as follows

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 def oracle(eng, x, Dataset, n0, output): """ Marks the solutions by flipping the output qubit, Args: eng (MainEngine): Main compiler engine the algorithm is being run on. x (Qureg): n-qubit quantum register Grover search is run on. Dataset(list): The dataset. n0: The threshold. output (Qubit): Output qubit to flip in order to mark the solution. """ # The size relationship is represented by 0,1, which is set to 1 if greater than n0, otherwise 0. fun = *len(Dataset) fun = [x-n0 for x in Dataset] fun = np.maximum(fun, 0) fun = np.minimum(fun, 1) a = sum(fun) n = math.ceil(math.log2(len(Dataset))) while a>0: num = *n p = fun.tolist().index(1) fun[p] = 0 i = 0 a = a-1 while p/2 != 0: num[i] = p % 2 p = p//2 i = i+1 a1 = sum(num) num1 = num while a1>0: p = num1.index(1) a1 = a1-1 num1[p] = 0 X | x[p] with Control(eng, x): X | output a1 = sum(num) while a1>0: p = num.index(1) a1 = a1-1 num[p] = 0 X | x[p] 

HiQ implementation of searching for the maximum value of the quantum algorithm

The code for quantum algorithm searching for the maximum value is as follows:

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169  import math from projectq.ops import H, Z, X, Measure, All, Ph, Rz from projectq.meta import Loop, Compute, Uncompute, Control from projectq.backends import CircuitDrawer, CommandPrinter from projectq.cengines import (MainEngine, AutoReplacer, LocalOptimizer, TagRemover, DecompositionRuleSet) import projectq.setups.decompositions from hiq.projectq.backends import SimulatorMPI from hiq.projectq.cengines import GreedyScheduler, HiQMainEngine from mpi4py import MPI def run_exactgrover(eng, n, oracle, oracle_modified): """ Runs exact Grover's algorithm on n qubit using the two kind of provided quantum oracles (oracle and oracle_modified). This is an algorithm which can output solution with probability 1. Args: eng (MainEngine): Main compiler engine to run Grover on. n (int): Number of bits in the solution. oracle (function): Function accepting the engine, an n-qubit register, and an output qubit which is flipped by the oracle for the correct bit string. Returns: solution (list): Solution bit-string. """ x = eng.allocate_qureg(n) # start in uniform superposition All(H) | x # number of iterations we have to run: num_it = int(math.pi/4.*math.sqrt(1 << n)) #phi is the parameter of modified oracle #varphi is the parameter of reflection across uniform superposition theta=math.asin(math.sqrt(1/(1 << n))) phi=math.acos(-math.cos(2*theta)/(math.sin(2*theta)*math.tan((2*num_it+1)*theta))) varphi=math.atan(1/(math.sin(2*theta)*math.sin(phi)*math.tan((2*num_it+1)*theta)))*2 # prepare the oracle output qubit (the one that is flipped to indicate the # solution. start in state 1/sqrt(2) * (|0> - |1>) s.t. a bit-flip turns # into a (-1)-phase. oracle_out = eng.allocate_qubit() X | oracle_out H | oracle_out # run num_it iterations with Loop(eng, num_it): # oracle adds a (-1)-phase to the solution oracle(eng, x, oracle_out) # reflection across uniform superposition with Compute(eng): All(H) | x All(X) | x with Control(eng, x[0:-1]): Z | x[-1] Uncompute(eng) # prepare the oracle output qubit (the one that is flipped to indicate the # solution. start in state |1> s.t. a bit-flip turns into a e^(i*phi)-phase. H | oracle_out oracle_modified(eng, x, oracle_out, phi) with Compute(eng): All(H) | x All(X) | x with Control(eng, x[0:-1]): Rz(varphi) | x[-1] Ph(varphi/2) | x[-1] Uncompute(eng) All(Measure) | x Measure | oracle_out eng.flush() # return result return [int(qubit) for qubit in x] def alternating_bits_oracle(eng, qubits, output): """ Marks the solution string 0, 1, 0,...,0 by flipping the qubit, conditioned on qubits being equal to the alternating bit-string. Args: eng (MainEngine): Main compiler engine the algorithm is being run on. qubits (Qureg): n-qubit quantum register Grover search is run on. output (Qubit): Output qubit to flip in order to mark the solution. """ with Compute(eng): X | qubits with Control(eng, qubits): X | output Uncompute(eng) def alternating_bits_oracle_modified(eng, qubits, output, phi): """ Marks the solution string 0,1,0,... by applying phase gate to the output bits, conditioned on qubits being equal to the alternating bit-string. Args: eng (MainEngine): Main compiler engine the algorithm is being run on. qubits (Qureg): n-qubit quantum register Grover search is run on. output (Qubit): Output qubit to mark the solution by a phase gate with parameter phi. """ with Compute(eng): All(X) | qubits[1::2] with Control(eng, qubits): Rz(phi) | output Ph(phi/2) | output Uncompute(eng) if __name__ == "__main__": # create a main compiler engine with a simulator backend: backend = SimulatorMPI(gate_fusion=True, num_local_qubits=20) # backend = CircuitDrawer() # locations = {} # for i in range(module.nqubits): # locations[i] = i # backend.set_qubit_locations(locations) cache_depth = 10 rule_set = DecompositionRuleSet(modules=[projectq.setups.decompositions]) engines = [TagRemover(), LocalOptimizer(cache_depth), AutoReplacer(rule_set), TagRemover(), LocalOptimizer(cache_depth) #,CommandPrinter() , GreedyScheduler() ] eng = HiQMainEngine(backend, engines) # run Grover search to find a 7-bit solution print("=====================================================================") print("= This is the Grover algorithm demo") print("= Change the list to search by modifying alternating_bits_oracle") print("= The chosen list is: [0, 1, 0, ....]") print("= The second element is marked") print("= Calling ExactGrover algorithm to find the marked element now...") # run Grover search to find a 7-bit solution N = 12 mark_bits = run_exactgrover(eng, N, alternating_bits_oracle, alternating_bits_oracle_modified) found = False for i in range(len(mark_bits)): if mark_bits[i] == 0: print("= Marked element is found, it index is: {}".format(i)) found = True if not found: print("= Cannot found the marked element!") print("=====================================================================") 

References: