dlib C++ Library - quantum_computing.h

    // Copyright (C) 2008  Davis E. King (davis@dlib.net)
    // License: Boost Software License   See LICENSE.txt for the full license.
    #ifndef DLIB_QUANTUM_COMPUTINg_1_
    #define DLIB_QUANTUM_COMPUTINg_1_

    #include <complex>
    #include <cmath>
    #include "../matrix.h"
    #include "../rand.h"
    #include "../enable_if.h"
    #include "../algs.h"
    #include "quantum_computing_abstract.h"

    namespace dlib
    {

        template <typename T>
        struct gate_traits {};

        namespace qc_helpers
        {

        // ------------------------------------------------------------------------------------

        // This is a template to compute the value of 2^n at compile time
            template <long n>
            struct exp_2_n
            {
                COMPILE_TIME_ASSERT(0 <= n && n <= 30);
                static const long value = exp_2_n<n-1>::value*2;
            };

            template <>
            struct exp_2_n<0>
            {
                static const long value = 1;
            };

        // ------------------------------------------------------------------------------------

        }

        typedef std::complex<double> qc_scalar_type;

    // ----------------------------------------------------------------------------------------

        class quantum_register
        {
        public:

            quantum_register()
            {
                set_num_bits(1);
            }

            int num_bits (
            ) const
            {
                return num_bits_in_register;
            }

            void set_num_bits (
                int num_bits
            )
            {
                // make sure requires clause is not broken
                DLIB_CASSERT(1 <= num_bits && num_bits <= 30,
                    "\tvoid quantum_register::set_num_bits()"
                    << "\n\tinvalid arguments to this function"
                    << "\n\tnum_bits: " << num_bits
                    << "\n\tthis:     " << this
                    );

                num_bits_in_register = num_bits;

                unsigned long size = 1;
                for (int i = 0; i < num_bits; ++i)
                    size *= 2;

                state.set_size(size);

                zero_all_bits();
            }

            void zero_all_bits()
            {
                set_all_elements(state,0);
                state(0) = 1;
            }

            void append (
                const quantum_register& reg
            )
            {
                num_bits_in_register += reg.num_bits_in_register;
                state = tensor_product(state, reg.state);
            }

            template <typename rand_type>
            bool measure_bit (
                int bit,
                rand_type& rnd
            )
            {
                // make sure requires clause is not broken
                DLIB_CASSERT(0 <= bit && bit < num_bits(),
                    "\tbool quantum_register::measure_bit()"
                    << "\n\tinvalid arguments to this function"
                    << "\n\tbit:        " << bit
                    << "\n\tnum_bits(): " << num_bits()
                    << "\n\tthis:       " << this
                    );

                const bool value = (rnd.get_random_double() < probability_of_bit(bit));

                // Next we set all the states where this bit doesn't have the given value to 0

                // But first make a mask that selects our bit
                unsigned long mask = 1;
                for (int i = 0; i < bit; ++i)
                    mask <<= 1;

                // loop over all the elements in the state vector and zero out those that
                // conflict with the measurement we just made.
                for (long r = 0; r < state.nr(); ++r)
                {
                    const unsigned long field = r;
                    // if this state indicates that the bit should be set and it isn't
                    if ((field & mask) && !value)
                    {
                        state(r) = 0;
                    }
                    // else if this state indicates that the bit should not be set and it is
                    else if (!(field & mask) && value)
                    {
                        state(r) = 0;
                    }
                }

                // normalize the state
                state = state/(std::sqrt(sum(norm(state))));

                return value;
            }

            template <typename rand_type>
            bool measure_and_remove_bit (
                int bit,
                rand_type& rnd
            )
            {
                // make sure requires clause is not broken
                DLIB_CASSERT(0 <= bit && bit < num_bits() && num_bits() > 0,
                    "\tbool quantum_register::measure_and_remove_bit()"
                    << "\n\tinvalid arguments to this function"
                    << "\n\tbit:        " << bit
                    << "\n\tnum_bits(): " << num_bits()
                    << "\n\tthis:       " << this
                    );

                const bool value = (rnd.get_random_double() < probability_of_bit(bit));
                quantum_register temp;
                temp.set_num_bits(num_bits()-1);

                // Next we set all the states where this bit doesn't have the given value to 0

                // But first make a mask that selects our bit
                unsigned long mask = 1;
                for (int i = 0; i < bit; ++i)
                    mask <<= 1;

                long count = 0;
                for (long r = 0; r < state.nr(); ++r)
                {
                    const unsigned long field = r;
                    // if this basis vector is one that matches the measured state then keep it
                    if (((field & mask) != 0) == value)
                    {
                        temp.state(count) = state(r);
                        ++count;
                    }
                }

                // normalize the state
                temp.state = temp.state/std::sqrt(sum(norm(temp.state)));

                temp.swap(*this);

                return value;
            }

            double probability_of_bit (
                int bit
            ) const
            {
                // make sure requires clause is not broken
                DLIB_CASSERT(0 <= bit && bit < num_bits(),
                    "\tdouble quantum_register::probability_of_bit()"
                    << "\n\tinvalid arguments to this function"
                    << "\n\tbit:        " << bit
                    << "\n\tnum_bits(): " << num_bits()
                    << "\n\tthis:       " << this
                    );

                // make a mask that selects our bit
                unsigned long mask = 1;
                for (int i = 0; i < bit; ++i)
                    mask <<= 1;

                // now find the total probability of all the states that have the given bit set
                double prob = 0;
                for (long r = 0; r < state.nr(); ++r)
                {
                    const unsigned long field = r;
                    if (field & mask)
                    {
                        prob += std::norm(state(r));
                    }
                }

                return prob;
            }

            const matrix<qc_scalar_type,0,1>& state_vector() const { return state; }
            matrix<qc_scalar_type,0,1>& state_vector() { return state; }

            void swap (
                quantum_register& item
            )
            {
                exchange(num_bits_in_register, item.num_bits_in_register);
                state.swap(item.state);
            }

        private:

            int num_bits_in_register;
            matrix<qc_scalar_type,0,1> state;
        };

        inline void swap (
            quantum_register& a,
            quantum_register& b
        ) { a.swap(b); }

    // ----------------------------------------------------------------------------------------

        template <typename T>
        class gate_exp
        {
        public:
            static const long num_bits = gate_traits<T>::num_bits;
            static const long dims = gate_traits<T>::dims;

            gate_exp(T& exp_) : exp(exp_) {}

            const qc_scalar_type operator() (long r, long c) const { return exp(r,c); }

            const matrix<qc_scalar_type> mat (
            ) const
            {
                matrix<qc_scalar_type,dims,dims> m;
                for (long r = 0; r < m.nr(); ++r)
                {
                    for (long c = 0; c < m.nc(); ++c)
                    {
                        m(r,c) = exp(r,c);
                    }
                }
                return m;
            }

            void apply_gate_to (quantum_register& reg) const
            {
                // make sure requires clause is not broken
                DLIB_CASSERT(reg.num_bits() == num_bits,
                    "\tvoid gate_exp::apply_gate_to()"
                    << "\n\tinvalid arguments to this function"
                    << "\n\treg.num_bits(): " << reg.num_bits()
                    << "\n\tnum_bits:       " << num_bits
                    << "\n\tthis:           " << this
                    );

                quantum_register temp(reg);

                // check if any of the elements of the register are 1 and if so then
                // we don't have to do the full matrix multiply.  Or check if only a small number are non-zero.
                long non_zero_elements = 0;
                for (long r = 0; r < dims; ++r)
                {
                    if (reg.state_vector()(r) != qc_scalar_type(0))
                        ++non_zero_elements;

                    reg.state_vector()(r) = 0;
                }

                if (non_zero_elements > 3)
                {
                    // do a full matrix multiply to compute the output state
                    for (long r = 0; r < dims; ++r)
                    {
                        reg.state_vector()(r) = compute_state_element(temp.state_vector(),r);
                    }
                }
                else
                {
                    // do a matrix multiply but only use the columns in the gate matrix
                    // that correspond to the non-zero register elements
                    for (long r = 0; r < dims; ++r)
                    {
                        if (temp.state_vector()(r) != qc_scalar_type(0))
                        {
                            for (long i = 0; i < dims; ++i)
                            {
                                reg.state_vector()(i) += temp.state_vector()(r)*exp(i,r);
                            }
                        }
                    }
                }
            }

            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 gate_exp::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
                    );

                return this->exp.compute_state_element(reg,row_idx);
            }

            const T& ref() const { return exp; }

        private:
            T& exp;
        };

    // ----------------------------------------------------------------------------------------

        template <typename T, typename U>
        class composite_gate;

        template <typename T, typename U>
        struct gate_traits<composite_gate<T,U> >
        {
            static const long num_bits = T::num_bits + U::num_bits;
            static const long dims = qc_helpers::exp_2_n<num_bits>::value;
        };

        template <typename T, typename U>
        class composite_gate : public gate_exp<composite_gate<T,U> >
        {
        public:

            typedef T lhs_type;
            typedef U rhs_type;

            composite_gate(const composite_gate& g) : gate_exp<composite_gate>(*this), lhs(g.lhs), rhs(g.rhs) {}

            composite_gate(
                const gate_exp<T>& lhs_,
                const gate_exp<U>& rhs_
            ) : gate_exp<composite_gate>(*this), lhs(lhs_.ref()), rhs(rhs_.ref()) {}

            static const long num_bits = gate_traits<composite_gate>::num_bits;
            static const long dims = gate_traits<composite_gate>::dims;

            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); }

            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 composite_gate::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
                    );

                qc_scalar_type result = 0;
                for (long c = 0; c < T::dims; ++c)
                {
                    if (lhs(row_idx/U::dims,c) != qc_scalar_type(0))
                    {
                        result += lhs(row_idx/U::dims,c) * rhs.compute_state_element(subm(reg,c*U::dims,0,U::dims,1), row_idx%U::dims);
                    }
                }

                return result;
            }

            const T lhs;
            const U rhs;
        };

    // ----------------------------------------------------------------------------------------

        template <long bits>
        class gate;
        template <long bits>
        struct gate_traits<gate<bits> >
        {
            static const long num_bits = bits;
            static const long dims = qc_helpers::exp_2_n<num_bits>::value;
        };

    // ----------------------------------------------------------------------------------------

        template <long bits>
        class gate : public gate_exp<gate<bits> >
        {
        public:
            gate() : gate_exp<gate>(*this) { set_all_elements(data,0); }
            gate(const gate& g) :gate_exp<gate>(*this), data(g.data) {}

            template <typename T>
            explicit gate(const gate_exp<T>& g) : gate_exp<gate>(*this)
            {
                COMPILE_TIME_ASSERT(T::num_bits == num_bits);
                for (long r = 0; r < dims; ++r)
                {
                    for (long c = 0; c < dims; ++c)
                    {
                        data(r,c) = g(r,c);
                    }
                }
            }

            static const long num_bits = gate_traits<gate>::num_bits;
            static const long dims = gate_traits<gate>::dims;

            const qc_scalar_type& operator() (long r, long c) const { return data(r,c); }
            qc_scalar_type& operator() (long r, long c)  { return data(r,c); }

            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 gate::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
                    );

                return (data*reg)(row_idx);
            }

        private:

            matrix<qc_scalar_type,dims,dims> data;
        };

    // ----------------------------------------------------------------------------------------

        namespace qc_helpers
        {
            // This is the maximum number of bits used for cached sub-matrices in composite_gate expressions
            const int qc_block_chunking_size = 8;

            template <typename T>
            struct is_composite_gate { const static bool value = false; };
            template <typename T, typename U>
            struct is_composite_gate<composite_gate<T,U> > { const static bool value = true; };

            // These overloads all deal with intelligently composing chains of composite_gate expressions
            // such that the resulting expression has the form:
            //    (gate_exp,(gate_exp,(gate_exp,(gate_exp()))))
            // and each gate_exp contains a cached gate matrix for a gate of at most qc_block_chunking_size bits.
            // This facilitates the optimizations in the compute_state_element() function.
            template <typename T, typename U, typename V, typename enabled = void>
            struct combine_gates;

            // This is a base case of this recursive template.  It takes care of converting small composite_gates into
            // cached gate objects.
            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)>::type >
            {
                typedef composite_gate<gate<T::num_bits + U::num_bits>,V>  result_type;

                static const result_type eval (
                    const composite_gate<T,U>& lhs,
                    const gate_exp<V>& rhs
                )
                {
                    typedef gate<T::num_bits + U::num_bits> gate_type;
                    return composite_gate<gate_type,V>(gate_type(lhs), rhs);
                }
            };

            // this is the recursive step of this template
            template <typename T, typename U, typename V>
            struct combine_gates<T,U,V,typename enable_if_c<(is_composite_gate<U>::value == true)>::type >
            {
                typedef typename combine_gates<typename U::lhs_type, typename U::rhs_type, V>::result_type inner_type;
                typedef composite_gate<T,inner_type> result_type;

                static const result_type eval (
                    const composite_gate<T,U>& lhs,
                    const gate_exp<V>& rhs
                )
                {
                    return composite_gate<T,inner_type>(lhs.lhs, combine_gates<typename U::lhs_type, typename U::rhs_type, V>::eval(lhs.rhs,rhs));
                }

            };

            // This is a base case of this recursive template.  It takes care of adding new gates when the left
            // 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_