coco
864 lines
27 KiB
C++
864 lines
27 KiB
C++
// Copyright (C) 2008 Davis E. King (davis@dlib.net)
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// License: Boost Software License See LICENSE.txt for the full license.
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#ifndef DLIB_QUANTUM_COMPUTINg_1_
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#define DLIB_QUANTUM_COMPUTINg_1_
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#include <complex>
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#include <cmath>
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#include "../matrix.h"
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#include "../rand.h"
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#include "../enable_if.h"
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#include "../algs.h"
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#include "quantum_computing_abstract.h"
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namespace dlib
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{
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template <typename T>
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struct gate_traits {};
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namespace qc_helpers
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{
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// ------------------------------------------------------------------------------------
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// This is a template to compute the value of 2^n at compile time
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template <long n>
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struct exp_2_n
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{
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COMPILE_TIME_ASSERT(0 <= n && n <= 30);
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static const long value = exp_2_n<n-1>::value*2;
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};
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template <>
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struct exp_2_n<0>
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{
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static const long value = 1;
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};
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// ------------------------------------------------------------------------------------
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}
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typedef std::complex<double> qc_scalar_type;
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// ----------------------------------------------------------------------------------------
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class quantum_register
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{
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public:
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quantum_register()
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{
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set_num_bits(1);
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}
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int num_bits (
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) const
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{
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return num_bits_in_register;
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}
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void set_num_bits (
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int num_bits
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)
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{
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// make sure requires clause is not broken
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DLIB_CASSERT(1 <= num_bits && num_bits <= 30,
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"\tvoid quantum_register::set_num_bits()"
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<< "\n\tinvalid arguments to this function"
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<< "\n\tnum_bits: " << num_bits
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<< "\n\tthis: " << this
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);
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num_bits_in_register = num_bits;
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unsigned long size = 1;
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for (int i = 0; i < num_bits; ++i)
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size *= 2;
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state.set_size(size);
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zero_all_bits();
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}
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void zero_all_bits()
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{
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set_all_elements(state,0);
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state(0) = 1;
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}
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void append (
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const quantum_register& reg
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)
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{
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num_bits_in_register += reg.num_bits_in_register;
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state = tensor_product(state, reg.state);
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}
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template <typename rand_type>
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bool measure_bit (
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int bit,
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rand_type& rnd
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)
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{
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// make sure requires clause is not broken
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DLIB_CASSERT(0 <= bit && bit < num_bits(),
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"\tbool quantum_register::measure_bit()"
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<< "\n\tinvalid arguments to this function"
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<< "\n\tbit: " << bit
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<< "\n\tnum_bits(): " << num_bits()
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<< "\n\tthis: " << this
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);
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const bool value = (rnd.get_random_double() < probability_of_bit(bit));
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// Next we set all the states where this bit doesn't have the given value to 0
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// But first make a mask that selects our bit
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unsigned long mask = 1;
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for (int i = 0; i < bit; ++i)
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mask <<= 1;
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// loop over all the elements in the state vector and zero out those that
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// conflict with the measurement we just made.
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for (long r = 0; r < state.nr(); ++r)
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{
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const unsigned long field = r;
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// if this state indicates that the bit should be set and it isn't
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if ((field & mask) && !value)
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{
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state(r) = 0;
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}
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// else if this state indicates that the bit should not be set and it is
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else if (!(field & mask) && value)
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{
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state(r) = 0;
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}
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}
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// normalize the state
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state = state/(std::sqrt(sum(norm(state))));
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return value;
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}
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template <typename rand_type>
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bool measure_and_remove_bit (
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int bit,
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rand_type& rnd
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)
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{
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// make sure requires clause is not broken
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DLIB_CASSERT(0 <= bit && bit < num_bits() && num_bits() > 0,
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"\tbool quantum_register::measure_and_remove_bit()"
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<< "\n\tinvalid arguments to this function"
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<< "\n\tbit: " << bit
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<< "\n\tnum_bits(): " << num_bits()
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<< "\n\tthis: " << this
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);
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const bool value = (rnd.get_random_double() < probability_of_bit(bit));
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quantum_register temp;
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temp.set_num_bits(num_bits()-1);
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// Next we set all the states where this bit doesn't have the given value to 0
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// But first make a mask that selects our bit
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unsigned long mask = 1;
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for (int i = 0; i < bit; ++i)
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mask <<= 1;
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long count = 0;
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for (long r = 0; r < state.nr(); ++r)
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{
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const unsigned long field = r;
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// if this basis vector is one that matches the measured state then keep it
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if (((field & mask) != 0) == value)
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{
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temp.state(count) = state(r);
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++count;
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}
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}
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// normalize the state
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temp.state = temp.state/std::sqrt(sum(norm(temp.state)));
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temp.swap(*this);
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return value;
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}
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double probability_of_bit (
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int bit
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) const
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{
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// make sure requires clause is not broken
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DLIB_CASSERT(0 <= bit && bit < num_bits(),
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"\tdouble quantum_register::probability_of_bit()"
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<< "\n\tinvalid arguments to this function"
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<< "\n\tbit: " << bit
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<< "\n\tnum_bits(): " << num_bits()
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<< "\n\tthis: " << this
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);
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// make a mask that selects our bit
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unsigned long mask = 1;
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for (int i = 0; i < bit; ++i)
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mask <<= 1;
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// now find the total probability of all the states that have the given bit set
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double prob = 0;
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for (long r = 0; r < state.nr(); ++r)
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{
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const unsigned long field = r;
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if (field & mask)
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{
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prob += std::norm(state(r));
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}
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}
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return prob;
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}
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const matrix<qc_scalar_type,0,1>& state_vector() const { return state; }
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matrix<qc_scalar_type,0,1>& state_vector() { return state; }
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void swap (
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quantum_register& item
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)
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{
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exchange(num_bits_in_register, item.num_bits_in_register);
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state.swap(item.state);
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}
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private:
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int num_bits_in_register;
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matrix<qc_scalar_type,0,1> state;
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};
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inline void swap (
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quantum_register& a,
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quantum_register& b
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) { a.swap(b); }
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// ----------------------------------------------------------------------------------------
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template <typename T>
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class gate_exp
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{
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public:
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static const long num_bits = gate_traits<T>::num_bits;
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static const long dims = gate_traits<T>::dims;
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gate_exp(T& exp_) : exp(exp_) {}
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const qc_scalar_type operator() (long r, long c) const { return exp(r,c); }
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const matrix<qc_scalar_type> mat (
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) const
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{
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matrix<qc_scalar_type,dims,dims> m;
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for (long r = 0; r < m.nr(); ++r)
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{
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for (long c = 0; c < m.nc(); ++c)
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{
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m(r,c) = exp(r,c);
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}
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}
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return m;
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}
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void apply_gate_to (quantum_register& reg) const
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{
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// make sure requires clause is not broken
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DLIB_CASSERT(reg.num_bits() == num_bits,
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"\tvoid gate_exp::apply_gate_to()"
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<< "\n\tinvalid arguments to this function"
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<< "\n\treg.num_bits(): " << reg.num_bits()
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<< "\n\tnum_bits: " << num_bits
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<< "\n\tthis: " << this
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);
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quantum_register temp(reg);
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// check if any of the elements of the register are 1 and if so then
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// we don't have to do the full matrix multiply. Or check if only a small number are non-zero.
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long non_zero_elements = 0;
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for (long r = 0; r < dims; ++r)
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{
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if (reg.state_vector()(r) != qc_scalar_type(0))
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++non_zero_elements;
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reg.state_vector()(r) = 0;
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}
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if (non_zero_elements > 3)
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{
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// do a full matrix multiply to compute the output state
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for (long r = 0; r < dims; ++r)
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{
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reg.state_vector()(r) = compute_state_element(temp.state_vector(),r);
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}
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}
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else
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{
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// do a matrix multiply but only use the columns in the gate matrix
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// that correspond to the non-zero register elements
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for (long r = 0; r < dims; ++r)
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{
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if (temp.state_vector()(r) != qc_scalar_type(0))
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{
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for (long i = 0; i < dims; ++i)
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{
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reg.state_vector()(i) += temp.state_vector()(r)*exp(i,r);
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}
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}
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}
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}
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}
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template <typename exp>
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qc_scalar_type compute_state_element (
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const matrix_exp<exp>& reg,
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long row_idx
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) const
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{
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// make sure requires clause is not broken
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DLIB_ASSERT(reg.nr() == dims && reg.nc() == 1 &&
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0 <= row_idx && row_idx < dims,
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"\tqc_scalar_type gate_exp::compute_state_element(reg,row_idx)"
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<< "\n\tinvalid arguments to this function"
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<< "\n\treg.nr(): " << reg.nr()
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<< "\n\treg.nc(): " << reg.nc()
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<< "\n\tdims: " << dims
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<< "\n\trow_idx: " << row_idx
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<< "\n\tthis: " << this
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);
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return this->exp.compute_state_element(reg,row_idx);
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}
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const T& ref() const { return exp; }
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private:
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T& exp;
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};
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// ----------------------------------------------------------------------------------------
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template <typename T, typename U>
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class composite_gate;
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template <typename T, typename U>
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struct gate_traits<composite_gate<T,U> >
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{
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static const long num_bits = T::num_bits + U::num_bits;
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static const long dims = qc_helpers::exp_2_n<num_bits>::value;
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};
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template <typename T, typename U>
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class composite_gate : public gate_exp<composite_gate<T,U> >
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{
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public:
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typedef T lhs_type;
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typedef U rhs_type;
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composite_gate(const composite_gate& g) : gate_exp<composite_gate>(*this), lhs(g.lhs), rhs(g.rhs) {}
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composite_gate(
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const gate_exp<T>& lhs_,
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const gate_exp<U>& rhs_
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) : gate_exp<composite_gate>(*this), lhs(lhs_.ref()), rhs(rhs_.ref()) {}
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static const long num_bits = gate_traits<composite_gate>::num_bits;
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static const long dims = gate_traits<composite_gate>::dims;
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const qc_scalar_type operator() (long r, long c) const { return lhs(r/U::dims,c/U::dims)*rhs(r%U::dims, c%U::dims); }
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template <typename exp>
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qc_scalar_type compute_state_element (
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const matrix_exp<exp>& reg,
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long row_idx
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) const
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{
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// make sure requires clause is not broken
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DLIB_ASSERT(reg.nr() == dims && reg.nc() == 1 &&
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0 <= row_idx && row_idx < dims,
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"\tqc_scalar_type composite_gate::compute_state_element(reg,row_idx)"
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<< "\n\tinvalid arguments to this function"
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<< "\n\treg.nr(): " << reg.nr()
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<< "\n\treg.nc(): " << reg.nc()
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<< "\n\tdims: " << dims
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<< "\n\trow_idx: " << row_idx
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<< "\n\tthis: " << this
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);
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qc_scalar_type result = 0;
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for (long c = 0; c < T::dims; ++c)
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{
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if (lhs(row_idx/U::dims,c) != qc_scalar_type(0))
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{
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result += lhs(row_idx/U::dims,c) * rhs.compute_state_element(subm(reg,c*U::dims,0,U::dims,1), row_idx%U::dims);
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}
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}
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return result;
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}
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const T lhs;
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const U rhs;
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};
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// ----------------------------------------------------------------------------------------
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template <long bits>
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class gate;
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template <long bits>
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struct gate_traits<gate<bits> >
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{
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static const long num_bits = bits;
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static const long dims = qc_helpers::exp_2_n<num_bits>::value;
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};
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// ----------------------------------------------------------------------------------------
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template <long bits>
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class gate : public gate_exp<gate<bits> >
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{
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public:
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gate() : gate_exp<gate>(*this) { set_all_elements(data,0); }
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gate(const gate& g) :gate_exp<gate>(*this), data(g.data) {}
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template <typename T>
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explicit gate(const gate_exp<T>& g) : gate_exp<gate>(*this)
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{
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COMPILE_TIME_ASSERT(T::num_bits == num_bits);
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for (long r = 0; r < dims; ++r)
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{
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for (long c = 0; c < dims; ++c)
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{
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data(r,c) = g(r,c);
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}
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}
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}
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static const long num_bits = gate_traits<gate>::num_bits;
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static const long dims = gate_traits<gate>::dims;
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const qc_scalar_type& operator() (long r, long c) const { return data(r,c); }
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qc_scalar_type& operator() (long r, long c) { return data(r,c); }
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template <typename exp>
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qc_scalar_type compute_state_element (
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const matrix_exp<exp>& reg,
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long row_idx
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) const
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{
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// make sure requires clause is not broken
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DLIB_ASSERT(reg.nr() == dims && reg.nc() == 1 &&
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0 <= row_idx && row_idx < dims,
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"\tqc_scalar_type gate::compute_state_element(reg,row_idx)"
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<< "\n\tinvalid arguments to this function"
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<< "\n\treg.nr(): " << reg.nr()
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<< "\n\treg.nc(): " << reg.nc()
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<< "\n\tdims: " << dims
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<< "\n\trow_idx: " << row_idx
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<< "\n\tthis: " << this
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);
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return (data*reg)(row_idx);
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}
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private:
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matrix<qc_scalar_type,dims,dims> data;
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};
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// ----------------------------------------------------------------------------------------
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namespace qc_helpers
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{
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// This is the maximum number of bits used for cached sub-matrices in composite_gate expressions
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const int qc_block_chunking_size = 8;
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template <typename T>
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struct is_composite_gate { const static bool value = false; };
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template <typename T, typename U>
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struct is_composite_gate<composite_gate<T,U> > { const static bool value = true; };
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// These overloads all deal with intelligently composing chains of composite_gate expressions
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// such that the resulting expression has the form:
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// (gate_exp,(gate_exp,(gate_exp,(gate_exp()))))
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// and each gate_exp contains a cached gate matrix for a gate of at most qc_block_chunking_size bits.
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// This facilitates the optimizations in the compute_state_element() function.
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template <typename T, typename U, typename V, typename enabled = void>
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struct combine_gates;
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// This is a base case of this recursive template. It takes care of converting small composite_gates into
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// cached gate objects.
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template <typename T, typename U, typename V>
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struct combine_gates<T,U,V,typename enable_if_c<(T::num_bits + U::num_bits <= qc_block_chunking_size)>::type >
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{
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typedef composite_gate<gate<T::num_bits + U::num_bits>,V> result_type;
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static const result_type eval (
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const composite_gate<T,U>& lhs,
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const gate_exp<V>& rhs
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)
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{
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typedef gate<T::num_bits + U::num_bits> gate_type;
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return composite_gate<gate_type,V>(gate_type(lhs), rhs);
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}
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};
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// this is the recursive step of this template
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template <typename T, typename U, typename V>
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struct combine_gates<T,U,V,typename enable_if_c<(is_composite_gate<U>::value == true)>::type >
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{
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typedef typename combine_gates<typename U::lhs_type, typename U::rhs_type, V>::result_type inner_type;
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typedef composite_gate<T,inner_type> result_type;
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static const result_type eval (
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const composite_gate<T,U>& lhs,
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const gate_exp<V>& rhs
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)
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{
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return composite_gate<T,inner_type>(lhs.lhs, combine_gates<typename U::lhs_type, typename U::rhs_type, V>::eval(lhs.rhs,rhs));
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}
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};
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// This is a base case of this recursive template. It takes care of adding new gates when the left
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// hand side is too big to just turn it into a cached gate object.
|
|
template <typename T, typename U, typename V>
|
|
struct combine_gates<T,U,V,typename enable_if_c<(T::num_bits + U::num_bits > qc_block_chunking_size &&
|
|
is_composite_gate<U>::value == false)>::type >
|
|
{
|
|
typedef composite_gate<T,composite_gate<U, V> > result_type;
|
|
|
|
static const result_type eval (
|
|
const composite_gate<T,U>& lhs,
|
|
const gate_exp<V>& rhs
|
|
)
|
|
{
|
|
return result_type(lhs.lhs, composite_gate<U,V>(lhs.rhs, rhs));
|
|
}
|
|
|
|
};
|
|
|
|
}
|
|
|
|
template <typename T, typename U>
|
|
const composite_gate<T,U> operator, (
|
|
const gate_exp<T>& lhs,
|
|
const gate_exp<U>& rhs
|
|
)
|
|
{
|
|
return composite_gate<T,U>(lhs,rhs);
|
|
}
|
|
|
|
template <typename T, typename U, typename V>
|
|
const typename qc_helpers::combine_gates<T,U,V>::result_type operator, (
|
|
const composite_gate<T,U>& lhs,
|
|
const gate_exp<V>& rhs
|
|
)
|
|
{
|
|
return qc_helpers::combine_gates<T,U,V>::eval(lhs,rhs);
|
|
}
|
|
|
|
// If you are getting an error here then it means that you are trying to combine a gate expression
|
|
// with an integer somewhere (and that is an error).
|
|
template <typename T> void operator, ( const gate_exp<T>&, int) { COMPILE_TIME_ASSERT(sizeof(T) > 100000000); }
|
|
template <typename T> void operator, ( int, const gate_exp<T>&) { COMPILE_TIME_ASSERT(sizeof(T) > 100000000); }
|
|
|
|
// ----------------------------------------------------------------------------------------
|
|
|
|
namespace quantum_gates
|
|
{
|
|
template <int control_bit, int target_bit>
|
|
class cnot;
|
|
|
|
template <int control_bit1, int control_bit2, int target_bit>
|
|
class toffoli;
|
|
}
|
|
|
|
template <int control_bit, int target_bit>
|
|
struct gate_traits<quantum_gates::cnot<control_bit, target_bit> >
|
|
{
|
|
static const long num_bits = tabs<control_bit-target_bit>::value+1;
|
|
static const long dims = qc_helpers::exp_2_n<num_bits>::value;
|
|
};
|
|
|
|
template <int control_bit1, int control_bit2, int target_bit>
|
|
struct gate_traits<quantum_gates::toffoli<control_bit1, control_bit2, target_bit> >
|
|
{
|
|
static const long num_bits = tmax<tabs<control_bit1-target_bit>::value,
|
|
tabs<control_bit2-target_bit>::value>::value+1;
|
|
static const long dims = qc_helpers::exp_2_n<num_bits>::value;
|
|
};
|
|
|
|
|
|
// ----------------------------------------------------------------------------------------
|
|
|
|
namespace quantum_gates
|
|
{
|
|
|
|
inline const gate<1> hadamard(
|
|
)
|
|
{
|
|
gate<1> h;
|
|
h(0,0) = std::sqrt(1/2.0);
|
|
h(0,1) = std::sqrt(1/2.0);
|
|
h(1,0) = std::sqrt(1/2.0);
|
|
h(1,1) = -std::sqrt(1/2.0);
|
|
return h;
|
|
}
|
|
|
|
// ------------------------------------------------------------------------------------
|
|
|
|
inline const gate<1> x(
|
|
)
|
|
{
|
|
gate<1> x;
|
|
x(0,1) = 1;
|
|
x(1,0) = 1;
|
|
return x;
|
|
}
|
|
|
|
// ------------------------------------------------------------------------------------
|
|
|
|
inline const gate<1> y(
|
|
)
|
|
{
|
|
gate<1> x;
|
|
qc_scalar_type i(0,1);
|
|
x(0,1) = -i;
|
|
x(1,0) = i;
|
|
return x;
|
|
}
|
|
|
|
// ------------------------------------------------------------------------------------
|
|
|
|
inline const gate<1> z(
|
|
)
|
|
{
|
|
gate<1> z;
|
|
z(0,0) = 1;
|
|
z(1,1) = -1;
|
|
return z;
|
|
}
|
|
|
|
// ------------------------------------------------------------------------------------
|
|
|
|
inline const gate<1> noop(
|
|
)
|
|
{
|
|
gate<1> i;
|
|
i(0,0) = 1;
|
|
i(1,1) = 1;
|
|
return i;
|
|
}
|
|
|
|
// ------------------------------------------------------------------------------------
|
|
|
|
template <int control_bit, int target_bit>
|
|
class cnot : public gate_exp<cnot<control_bit, target_bit> >
|
|
{
|
|
public:
|
|
COMPILE_TIME_ASSERT(control_bit != target_bit);
|
|
|
|
cnot() : gate_exp<cnot>(*this)
|
|
{
|
|
const int min_bit = std::min(control_bit, target_bit);
|
|
|
|
control_mask = 1;
|
|
target_mask = 1;
|
|
|
|
// make the masks so that their only on bit corresponds to the given control_bit and target_bit bits
|
|
for (int i = 0; i < control_bit-min_bit; ++i)
|
|
control_mask <<= 1;
|
|
for (int i = 0; i < target_bit-min_bit; ++i)
|
|
target_mask <<= 1;
|
|
}
|
|
|
|
static const long num_bits = gate_traits<cnot>::num_bits;
|
|
static const long dims = gate_traits<cnot>::dims;
|
|
|
|
const qc_scalar_type operator() (long r, long c) const
|
|
{
|
|
unsigned long output;
|
|
// if the input control bit is set
|
|
if (control_mask&c)
|
|
{
|
|
output = c^target_mask;
|
|
}
|
|
else
|
|
{
|
|
output = c;
|
|
}
|
|
|
|
if ((unsigned long)r == output)
|
|
return 1;
|
|
else
|
|
return 0;
|
|
}
|
|
|
|
template <typename exp>
|
|
qc_scalar_type compute_state_element (
|
|
const matrix_exp<exp>& reg,
|
|
long row_idx
|
|
) const
|
|
{
|
|
// make sure requires clause is not broken
|
|
DLIB_ASSERT(reg.nr() == dims && reg.nc() == 1 &&
|
|
0 <= row_idx && row_idx < dims,
|
|
"\tqc_scalar_type cnot::compute_state_element(reg,row_idx)"
|
|
<< "\n\tinvalid arguments to this function"
|
|
<< "\n\treg.nr(): " << reg.nr()
|
|
<< "\n\treg.nc(): " << reg.nc()
|
|
<< "\n\tdims: " << dims
|
|
<< "\n\trow_idx: " << row_idx
|
|
<< "\n\tthis: " << this
|
|
);
|
|
|
|
|
|
unsigned long output = row_idx;
|
|
// if the input control bit is set
|
|
if (control_mask&output)
|
|
{
|
|
output = output^target_mask;
|
|
}
|
|
|
|
return reg(output);
|
|
}
|
|
|
|
private:
|
|
|
|
unsigned long control_mask;
|
|
unsigned long target_mask;
|
|
|
|
|
|
};
|
|
|
|
// ------------------------------------------------------------------------------------
|
|
|
|
template <int control_bit1, int control_bit2, int target_bit>
|
|
class toffoli : public gate_exp<toffoli<control_bit1, control_bit2, target_bit> >
|
|
{
|
|
public:
|
|
COMPILE_TIME_ASSERT(control_bit1 != target_bit && control_bit2 != target_bit && control_bit1 != control_bit2);
|
|
COMPILE_TIME_ASSERT((control_bit1 < target_bit && control_bit2 < target_bit) ||(control_bit1 > target_bit && control_bit2 > target_bit) );
|
|
|
|
toffoli() : gate_exp<toffoli>(*this)
|
|
{
|
|
const int min_bit = std::min(std::min(control_bit1, control_bit2), target_bit);
|
|
|
|
control1_mask = 1;
|
|
control2_mask = 1;
|
|
target_mask = 1;
|
|
|
|
// make the masks so that their only on bit corresponds to the given control_bit1 and target_bit bits
|
|
for (int i = 0; i < control_bit1-min_bit; ++i)
|
|
control1_mask <<= 1;
|
|
for (int i = 0; i < control_bit2-min_bit; ++i)
|
|
control2_mask <<= 1;
|
|
for (int i = 0; i < target_bit-min_bit; ++i)
|
|
target_mask <<= 1;
|
|
}
|
|
|
|
static const long num_bits = gate_traits<toffoli>::num_bits;
|
|
static const long dims = gate_traits<toffoli>::dims;
|
|
|
|
const qc_scalar_type operator() (long r, long c) const
|
|
{
|
|
unsigned long output;
|
|
// if the input control bits are set
|
|
if ((control1_mask&c) && (control2_mask&c))
|
|
{
|
|
output = c^target_mask;
|
|
}
|
|
else
|
|
{
|
|
output = c;
|
|
}
|
|
|
|
if ((unsigned long)r == output)
|
|
return 1;
|
|
else
|
|
return 0;
|
|
}
|
|
|
|
template <typename exp>
|
|
qc_scalar_type compute_state_element (
|
|
const matrix_exp<exp>& reg,
|
|
long row_idx
|
|
) const
|
|
{
|
|
// make sure requires clause is not broken
|
|
DLIB_ASSERT(reg.nr() == dims && reg.nc() == 1 &&
|
|
0 <= row_idx && row_idx < dims,
|
|
"\tqc_scalar_type toffoli::compute_state_element(reg,row_idx)"
|
|
<< "\n\tinvalid arguments to this function"
|
|
<< "\n\treg.nr(): " << reg.nr()
|
|
<< "\n\treg.nc(): " << reg.nc()
|
|
<< "\n\tdims: " << dims
|
|
<< "\n\trow_idx: " << row_idx
|
|
<< "\n\tthis: " << this
|
|
);
|
|
|
|
|
|
unsigned long output;
|
|
// if the input control bits are set
|
|
if ((control1_mask&row_idx) && (control2_mask&row_idx))
|
|
{
|
|
output = row_idx^target_mask;
|
|
}
|
|
else
|
|
{
|
|
output = row_idx;
|
|
}
|
|
|
|
return reg(output);
|
|
|
|
}
|
|
|
|
private:
|
|
|
|
unsigned long control1_mask;
|
|
unsigned long control2_mask;
|
|
unsigned long target_mask;
|
|
|
|
|
|
};
|
|
|
|
|
|
// ------------------------------------------------------------------------------------
|
|
|
|
}
|
|
|
|
// ----------------------------------------------------------------------------------------
|
|
|
|
}
|
|
|
|
#endif // DLIB_QUANTUM_COMPUTINg_1_
|
|
|
|
|