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Dirac demo (quantum mechanics with C++). Please note: There is a problem with SlideShare updating documents, so there may be divergence between the code and the copied & pasted output.

Russell Childs, profile picture
Russell Childs

The document demonstrates a C++ framework for quantum mechanics using matrix mechanics with various quantum states, operators, and gates. It includes examples of operations with qubits, inner and outer products, and quantum circuits, specifically focusing on complex exponential forms and tensor products. Additionally, the document outlines a measurement process performed on quantum states by different entities, Alice and Bob.

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@copyright Russell John Childs, PhD, 11th March 2017 ======================================================================================== Simple demo of C++ framework allowing quantum mechanics to be done with Matrix Mechanics using Dirac Bras, Kets, inner, outer products, operators. ======================================================================================== Output of code (see later): =========================================== Usage model: =========================================== e^(i*theta): Polar notation cos theta + i sin theta a + i*b: Cartesian notation a + i b (e^(i*p/4))*Bra<3>{"010"}: Creates 3-qubit bra e^(i pi/4) <010| Ket<3>{"010"}: Creates 3-qubit ket |010> Bra<3>{"010"}*Ket<3>{"010"}: Inner product <010|010> = 1 Bra<3>{"010"}*Ket<3>{"110"}: Inner product <010|110> = 0 Bra<2>{"01"}*Ket<3>{"011"}: Inner product <01|011> = |1> Bra<2>{"01"}*Bra<3>{"011"}: tensor prod <01|<011| = <01011| Ket<2>{"01"}*Ket<3>{"011"}: tensor prod |01>|011> = |01011> cos(theta)*Ket<1>{"0"}*Bra<1>{"0"} - sin(theta)*Ket<1>{"0"}*Bra<1>{"1"}+ sin(theta)*Ket<1>{"1"}*Bra<1>{"0"}+ cos(theta)*Ket<1>{"1"}*Bra<1>{"1"}: Ry(2*theta) = -- -- | cos theta |0><0| -sin theta |0><1| | | sin theta |1><0| cos theta |1><1| | -- -- Passive rotation operator. Applied to alpha*Ket<1>{"0"} + beta*Ket<1>{"1"} = alpha |0> + beta |1> = -- -- | alpha | | beta | -- -- Ket<1>{"0"}*Bra<1>{"0"} & Ket<1>{"1"}*Bra<1>{"1"} - operator tensor product |0><0| & |1><1| = |01><01| (Ket<2>{"00"} >> Hadamard >> CNOT21) - the quantum circuit: |00> -- H -- CNOT21 = 1/sqrt_2(|00> + |11>), i.e. entangled (psi >> M >> bob) - performs a measurement on psi using the operator 'bob' =========================================== End of Usage model: =========================================== gates.X -> (1+i*0)|0><1| + (1+i*0)|1><0| gates.Y -> (0-i*1)|0><1| + (0+i*1)|1><0| gates.Z -> (1+i*0)|0><0| + (-1+i*0)|1><1| gates.H -> (4.32978e-17+i*0.707107)|0><0| + (4.32978e-17+i*0.707107)|0><1| + (-4.32978e-17+i*0.707107)|1><0| + (4.32978e- 17-i*0.707107)|1><1| gates.S -> (1+i*0)|0><0| + (0+i*1)|1><1| gates.T -> (1+i*0)|0><0| + (0.707107+i*0.707107)|1><1| CNOT = ((Ket<1>{"0"}*Bra<1>{"0"}&gates.I) + (Ket<1>{"1"}*Bra<1>{"1"}&gates.X)) -> (1+i*0)|00><00| + (1+i*0)|01><01| + (1+i*0)|10><11| + (1+i*0)|11><10| gates.controlled(gates.X) -> (1+i*0)|00><00| + (1+i*0)|01><01| + (1+i*0)|10><11| + (1+i*0)|11><10| ~((e^(i*pi/4))*Ket<1>{"0"}*Bra<1>{"1"}) = ~(e^ip/4 |0><1|) -> (0.707107-i*0.707107)|1><0| ~gates.S = ~((1+i*0)|0><0| + (0+i*1)|1><1|) -> (1+i*0)|0><0| + (0-i*1)|1><1| Psi<Dirac::K,1>({"0"}) = |0> -> (1+i*0)*|0> (Psi<Dirac::K,1>({"0"})>>gates.H) - single qubit quantum circuit |0> -- H -> (-4.32978e-17+i*0.707107)*|1> + (4.32978e- 17+i*0.707107)*|0> (Psi<Dirac::K,2>({"00"}) >> Hadamard) -> (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|11> + (-4.32978e- 17+i*0.707107)*|01> CNOT21 -> (1+i*0)|00><00| + (1+i*0)|01><11| + (1+i*0)|10><10| + (1+i*0)|11><01| (Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21) -> (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e- 17+i*0.707107)*|11> Bob -> (1+i*0)|00><00| + (1+i*0)|01><01| + (1+i*0)|10><10| + (1+i*0)|11><11| Alice -> (1+i*0)|00><00| + (1+i*0)|01><01| + (1+i*0)|10><10| + (1+i*0)|11><11| Measurements: psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (-6.12323e- 17+i*1)*|11>, Alice : (-6.12323e-17+i*1)*|11>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (-6.12323e- 17+i*1)*|11>, Alice : (-6.12323e-17+i*1)*|11>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (-6.12323e- 17+i*1)*|11>, Alice : (-6.12323e-17+i*1)*|11>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>}
psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (-6.12323e- 17+i*1)*|11>, Alice : (-6.12323e-17+i*1)*|11>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} (10+20*i)*K{"1010"} -> (10+i*20)*|1010> (10+20*i)*B{"1010"} -> <1010|*(10+i*20) (e^(i*pi/4))*B{"1010"}*K{"1010"}*~(e^(i*pi/4)) -> 1+i*0 (K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"})) -> (1+i*0)|1010><1010| (K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"}))*(K{"1010"}*~(e^(i*pi/4))) -> (0.707107-i*0.707107)*|1010> (B{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"})) -> <1010|*(0.707107-i*0.707107) (K{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))) -> (2.22045e-16-i*1)*|10101010> (B{"1010"}*~(e^(i*pi/4)))*(B{"1010"}*~(e^(i*pi/4))) -> <10101010|*(2.22045e-16-i*1) Bra<2>{"10"}*Ket<3>{"100"} -> (1+i*0)*|0> Bra<2>{"11"}*Ket<3>{"100"} -> (0+i*0)*|0> Bra<5>{"10011"}*Ket<3>{"100"} -> <11|*(1+i*0) (Bra<1>{"1"}*Bra<1>{"0"})*(Ket<1>{"1"}*Ket<1>{"0"}*Ket<1>{"0"}) -> (1+i*0)*|0> (Bra<1>{"1"}*Bra<1>{"1"})*(Ket<1>{"1"}*Ket<2>{"00"}) -> (0+i*0)*|0> Bra<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*(Bra<2>{"10"}*Bra<1>{"0"}) -> (1+i*0)|00><10011100| Ket<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*(Bra<2>{"10"}*Bra<1>{"0"}) -> (1+i*0)|1100><10011100| std::cos(pi/4)*Ket<1>{"0"} + std::sin(pi/4)*(e^(i*pi/4))*Ket<1>{"1"} ->(0.5+i*0.5)*|1> + (0.707107+i*0)*|0> ((elem*Ket<1>{"0"}*Bra<1>{"0"} - elem*Ket<1>{"0"}*Bra<1>{"1"} + (elem*Ket<1>{"1"}*Bra<1>{"0"} + elem*Ket<1>{"1"}*Bra<1>{"1"}))) -> (0.707107+i*0)|0><0| + (-0.707107+i*0)|0><1| + (0.707107+i*0)|1><0| + (0.707107+i*0)|1><1| 10*i -> +i*10, i*10 -> +i*10 (10*(-i))*(10*i) -> 100 abs(10*i) -> 10, abs(i*10) -> 10 10 +100*i -> 10+i*100, 2*(10 +100*i) -> 20+i*200, (10 +100*i)*3 -> 30+i*300, 4*C{10,100*i} -> 40+i*400, C{10,100*i}*5 -> 50+i*500 (2+3*i)*C{4,5*i}-> -7+i*22 ~(2+3*i)*C{2,3*i} -> 13+i*0 abs(~(2+3*i)*C{2,3*i}) -> 13 (e^(i*pi/4)).m_real -> 0.707107, abs((e^(i*pi/4)).m_real) -> 0.707107 (1000.0*e^(-i*pi/4)) -> 707.107-i*707.107, 1000*~(e^(i*pi/4)) -> 707.107-i*707.107 (e^(i*pi/4))*(e^(i*pi/4)) -> 2.22045e-16+i*1 ~(e^(i*pi/4))*(e^(i*pi/4)) -> 1+i*0 Code: int main(void) { using namespace QuantumGates; typedef QuantumGates::Ket<4> K; typedef QuantumGates::Bra<4> B; std::cout << "===========================================" << std::endl; std::cout << "Usage model:" << std::endl; std::cout << "===========================================" << std::endl; std::cout << "e^(i*theta): Polar notation cos theta + i sin theta" << std::endl; std::cout << "a + i*b: Cartesian notation a + i b" << std::endl; std::cout << "(e^(i*p/4))*Bra<3>{"010"}: Creates 3-qubit bra e^(i pi/4) " "<010|" << std::endl; std::cout << "Ket<3>{"010"}: Creates 3-qubit ket |010>" << std::endl; std::cout << "Bra<3>{"010"}*Ket<3>{"010"}: Inner product <010|010> = 1" << std::endl; std::cout << "Bra<3>{"010"}*Ket<3>{"110"}: Inner product <010|110> = 0" << std::endl; std::cout << "Bra<2>{"01"}*Ket<3>{"011"}: Inner product <01|011> = " "|1>" << std::endl; std::cout << "Bra<2>{"01"}*Bra<3>{"011"}: tensor prod <01|<011| " "= <01011|" << std::endl; std::cout << "Ket<2>{"01"}*Ket<3>{"011"}: tensor prod |01>|011> = " "|01011>" << std::endl; std::cout << "cos(theta)*Ket<1>{"0"}*Bra<1>{"0"} " "- sin(theta)*Ket<1>{"0"}*Bra<1>{"1"}" "+ sin(theta)*Ket<1>{"1"}*Bra<1>{"0"}" "+ cos(theta)*Ket<1>{"1"}*Bra<1>{"1"}: " << std::endl << " Ry(2*theta) = " << std::endl << " -- --" << std::endl<< " | cos theta |0><0| -sin theta |0><1| |"<< std::endl<< " | sin theta |1><0| cos theta |1><1| |"<< std::endl<< " -- --" << std::endl<<
" Passive rotation operator. Applied to" << std::endl << " alpha*Ket<1>{"0"} + beta*Ket<1>{"1"} " " = alpha |0> + beta |1> =" << std::endl<< " -- --" << std::endl<< " | alpha |" << std::endl<< " | beta |" << std::endl<< " -- --" << std::endl; std::cout << "Ket<1>{"0"}*Bra<1>{"0"} & Ket<1>{"1"}*Bra<1>{"1"} -" " operator tensor product |0><0| & |1><1| = |01><01|" << std::endl; std::cout << "(Ket<2>{"00"} >> Hadamard >> CNOT21) - the quantum circuit:" " |00> -- H -- CNOT21 = 1/sqrt_2(|00> + |11>), i.e. entangled" << std::endl; std::cout << "(psi >> M >> bob) - performs a measurement on psi " "using the operator 'bob' " << std::endl; std::cout << "===========================================" << std::endl; std::cout << "End of Usage model:" << std::endl; std::cout << "===========================================" << std::endl; auto gates = Gate::instance(); std::cout << "gates.X -> " << gates.X << std::endl; std::cout << "gates.Y -> " << gates.Y << std::endl; std::cout << "gates.Z -> " << gates.Z << std::endl; std::cout << "gates.H -> " << gates.H << std::endl; std::cout << "gates.S -> " << gates.S << std::endl; std::cout << "gates.T -> " << gates.T << std::endl; auto CNOT = ((Ket<1>{"0"}*Bra<1>{"0"}&gates.I) + (Ket<1>{"1"}*Bra<1>{"1"}&gates.X)); std::cout << "CNOT = ((Ket<1>{"0"}*Bra<1>{"0"}&gates.I) + " "(Ket<1>{"1"}*Bra<1>{"1"}&gates.X)) -> " << CNOT << std::endl; std::cout << "gates.controlled(gates.X) -> " << gates.controlled(gates.X) << std::endl; std::cout << "~((e^(i*pi/4))*Ket<1>{"0"}*Bra<1>{"1"}) " "= ~(e^ip/4 |0><1|) -> " << ~((e^(i*pi/4))*Ket<1>{"0"}*Bra<1>{1}) << std::endl; std::cout << "~gates.S = ~(" << gates.S << ") -> " << ~gates.S << std::endl; std::cout << "Psi<Dirac::K,1>({"0"}) = |0> -> " << Psi<Dirac::K,1>({"0"}) << std::endl; std::cout << "(Psi<Dirac::K,1>({"0"})>>gates.H) - single qubit quantum circuit" " |0> -- H -> " << (Psi<Dirac::K,1>({"0"})>>gates.H) << std::endl; auto Hadamard = ((Ket<1>{"0"}*Bra<1>{"0"}&gates.H) + (Ket<1>{"1"}*Bra<1>{"1"}&gates.I)); std::cout << "(Psi<Dirac::K,2>({"00"}) >> Hadamard) -> " << (Psi<Dirac::K,2>({"00"}) >> Hadamard) << std::endl; auto CNOT21 = Ket<2>{"00"}*Bra<2>{"00"} + Ket<2>{"01"}*Bra<2>{"11"} + Ket<2>{"10"}*Bra<2>{"10"} + Ket<2>{"11"}*Bra<2>{"01"}; std::cout << "CNOT21 -> "<< CNOT21 << std::endl; std::cout << "(Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21) -> " << (Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21) << std::endl; auto psi = (Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21); auto bob = ((Ket<1>{"0"}*Bra<1>{"0"} + Ket<1>{"1"}*Bra<1>{"1"}) & gates.I); auto alice = (gates.I &(Ket<1>{"0"}*Bra<1>{"0"} + Ket<1>{"1"}*Bra<1>{"1"})); std::cout << "Bob -> " << bob << std::endl; std::cout << "Alice -> " << alice << std::endl; std::cout << "Measurements:" << std::endl; for(unsigned i=0; i<10; ++i) { auto psi = (Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21); std::cout << "psi = " << psi << ", "; std::cout << "Bob : " << (psi >> M >> bob) << ", "; std::cout << "Alice : " << (psi >> M >> alice) << "} "; std::cout << std::endl; } std::cout << "(10+20*i)*K{"1010"} -> " << (10+20*i)*K{"1010"}<< std::endl; std::cout << "(10+20*i)*B{"1010"} -> " << (10+20*i)*B{"1010"}<< std::endl; std::cout << "(e^(i*pi/4))*B{"1010"}*K{"1010"}*~(e^(i*pi/4)) -> " << (e^(i*pi/4))*B{"1010"}*K{"1010"}*~(e^(i*pi/4)) << std::endl; std::cout << "(K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"})) -> " <<(K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"})) << std::endl; std::cout << "(K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"}))*"
"(K{"1010"}*~(e^(i*pi/4)))" " -> " << (K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"}))* (K{"1010"}*~(e^(i*pi/4))) << std::endl; std::cout << "(B{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))*" "((e^(i*pi/4))*B{"1010"})) -> " << (B{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))* ((e^(i*pi/4))*B{"1010"})) << std::endl; std::cout << "(K{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))) -> " << (K{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))) << std::endl; std::cout << "(B{"1010"}*~(e^(i*pi/4)))*(B{"1010"}*~(e^(i*pi/4))) -> " << (B{"1010"}*~(e^(i*pi/4)))*(B{"1010"}*~(e^(i*pi/4))) << std::endl; std::cout << "Bra<2>{"10"}*Ket<3>{"100"} -> " << Bra<2>{"10"}*Ket<3>{"100"} << std::endl; std::cout << "Bra<2>{"11"}*Ket<3>{"100"} -> " << Bra<2>{"11"}*Ket<3>{"100"} << std::endl; std::cout << "Bra<5>{"10011"}*Ket<3>{"100"} -> " << Bra<5>{"10011"}*Ket<3>{"100"} << std::endl; std::cout << "(Bra<1>{"1"}*Bra<1>{"0"})*(Ket<1>{"1"}*Ket<1>{"0"}*Ket<1>{"0"})" " -> " << (Bra<1>{"1"}*Bra<1>{"0"})*(Ket<1>{"1"}*Ket<1>{"0"}*Ket<1>{"0"}) << std::endl; std::cout << "(Bra<1>{"1"}*Bra<1>{"1"})*(Ket<1>{"1"}*Ket<2>{"00"}) " "-> " << (Bra<1>{"1"}*Bra<1>{"1"})*(Ket<1>{"1"}*Ket<2>{"00"}) << std::endl; std::cout << "Bra<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*" "(Bra<2>{"10"}*Bra<1>{"0"}) -> " << Bra<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*(Bra<2>{"10"}*Bra<1>{"0"}) << std::endl; std::cout << "Ket<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*" "(Bra<2>{"10"}*Bra<1>{"0"}) -> " << Ket<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})* (Bra<2>{"10"}*Bra<1>{"0"}) << std::endl; std::cout << "std::cos(pi/4)*Ket<1>{"0"} + std::sin(pi/4)*(e^(i*pi/4))*" "Ket<1>{"1"} ->" << std::cos(pi/4)*Ket<1>{"0"} + std::sin(pi/4)*(e^(i*pi/4))*Ket<1>{"1"} << std::endl; auto elem = 1/std::sqrt(2); std::cout <<"((elem*Ket<1>{"0"}*Bra<1>{"0"} " "- elem*Ket<1>{"0"}*Bra<1>{"1"} " "+ (elem*Ket<1>{"1"}*Bra<1>{"0"} " "+ elem*Ket<1>{"1"}*Bra<1>{"1"})))" " -> " << ((elem*Ket<1>{"0"}*Bra<1>{"0"} - elem*Ket<1>{"0"}*Bra<1>{"1"}+ (elem*Ket<1>{"1"}*Bra<1>{"0"} + elem*Ket<1>{"1"}*Bra<1>{"1"}))) << std::endl; std::cout << "10*i -> " << 10*i << ", i*10 -> " << i*10 << std::endl; std::cout << "(10*(-i))*(10*i) -> " << (10*(-i))*(10*i) << std::endl; std::cout << "abs(10*i) -> " << abs(10*i) << ", abs(i*10) -> " << abs(i*10)<< std::endl; std::cout << "10 +100*i -> " << 10 +100*i << ", 2*(10 +100*i) -> "<< 2*(10 +100*i) << ", (10 +100*i)*3 -> " << (10 +100*i)*3 << ", 4*C{10,100*i} -> " << 4*C{10,100*i} << ", C{10,100*i}*5 -> " << C{10,100*i}*5 << std::endl; std::cout << "(2+3*i)*C{4,5*i}-> " << (2+3*i)*C{4,5*i} << std::endl; std::cout << "~(2+3*i)*C{2,3*i} -> " << ~(2+3*i)*C{2,3*i} << std::endl; std::cout << "abs(~(2+3*i)*C{2,3*i}) -> " << abs(~(2+3*i)*C{2,3*i}) << std::endl; auto x = e^(i*pi/4); std::cout << "(e^(i*pi/4)).m_real -> " << x.m_real << ", abs((e^(i*pi/4)).m_real) -> " << abs(x.m_imag)<<std::endl; std::cout << "(1000.0*e^(-i*pi/4)) -> " << (1000.0*e^(-i*pi/4)) << ", 1000*~(e^(i*pi/4)) -> " << 1000*~(e^(i*pi/4)) << std::endl; std::cout << "(e^(i*pi/4))*(e^(i*pi/4)) -> " << x*x << std::endl; std::cout << "~(e^(i*pi/4))*(e^(i*pi/4)) -> "<< ~x*x << std::endl;
return 0; }

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Dirac demo (quantum mechanics with C++). Please note: There is a problem with SlideShare updating documents, so there may be divergence between the code and the copied & pasted output.

  • 1. @copyright Russell John Childs, PhD, 11th March 2017 ======================================================================================== Simple demo of C++ framework allowing quantum mechanics to be done with Matrix Mechanics using Dirac Bras, Kets, inner, outer products, operators. ======================================================================================== Output of code (see later): =========================================== Usage model: =========================================== e^(i*theta): Polar notation cos theta + i sin theta a + i*b: Cartesian notation a + i b (e^(i*p/4))*Bra<3>{"010"}: Creates 3-qubit bra e^(i pi/4) <010| Ket<3>{"010"}: Creates 3-qubit ket |010> Bra<3>{"010"}*Ket<3>{"010"}: Inner product <010|010> = 1 Bra<3>{"010"}*Ket<3>{"110"}: Inner product <010|110> = 0 Bra<2>{"01"}*Ket<3>{"011"}: Inner product <01|011> = |1> Bra<2>{"01"}*Bra<3>{"011"}: tensor prod <01|<011| = <01011| Ket<2>{"01"}*Ket<3>{"011"}: tensor prod |01>|011> = |01011> cos(theta)*Ket<1>{"0"}*Bra<1>{"0"} - sin(theta)*Ket<1>{"0"}*Bra<1>{"1"}+ sin(theta)*Ket<1>{"1"}*Bra<1>{"0"}+ cos(theta)*Ket<1>{"1"}*Bra<1>{"1"}: Ry(2*theta) = -- -- | cos theta |0><0| -sin theta |0><1| | | sin theta |1><0| cos theta |1><1| | -- -- Passive rotation operator. Applied to alpha*Ket<1>{"0"} + beta*Ket<1>{"1"} = alpha |0> + beta |1> = -- -- | alpha | | beta | -- -- Ket<1>{"0"}*Bra<1>{"0"} & Ket<1>{"1"}*Bra<1>{"1"} - operator tensor product |0><0| & |1><1| = |01><01| (Ket<2>{"00"} >> Hadamard >> CNOT21) - the quantum circuit: |00> -- H -- CNOT21 = 1/sqrt_2(|00> + |11>), i.e. entangled (psi >> M >> bob) - performs a measurement on psi using the operator 'bob' =========================================== End of Usage model: =========================================== gates.X -> (1+i*0)|0><1| + (1+i*0)|1><0| gates.Y -> (0-i*1)|0><1| + (0+i*1)|1><0| gates.Z -> (1+i*0)|0><0| + (-1+i*0)|1><1| gates.H -> (4.32978e-17+i*0.707107)|0><0| + (4.32978e-17+i*0.707107)|0><1| + (-4.32978e-17+i*0.707107)|1><0| + (4.32978e- 17-i*0.707107)|1><1| gates.S -> (1+i*0)|0><0| + (0+i*1)|1><1| gates.T -> (1+i*0)|0><0| + (0.707107+i*0.707107)|1><1| CNOT = ((Ket<1>{"0"}*Bra<1>{"0"}&gates.I) + (Ket<1>{"1"}*Bra<1>{"1"}&gates.X)) -> (1+i*0)|00><00| + (1+i*0)|01><01| + (1+i*0)|10><11| + (1+i*0)|11><10| gates.controlled(gates.X) -> (1+i*0)|00><00| + (1+i*0)|01><01| + (1+i*0)|10><11| + (1+i*0)|11><10| ~((e^(i*pi/4))*Ket<1>{"0"}*Bra<1>{"1"}) = ~(e^ip/4 |0><1|) -> (0.707107-i*0.707107)|1><0| ~gates.S = ~((1+i*0)|0><0| + (0+i*1)|1><1|) -> (1+i*0)|0><0| + (0-i*1)|1><1| Psi<Dirac::K,1>({"0"}) = |0> -> (1+i*0)*|0> (Psi<Dirac::K,1>({"0"})>>gates.H) - single qubit quantum circuit |0> -- H -> (-4.32978e-17+i*0.707107)*|1> + (4.32978e- 17+i*0.707107)*|0> (Psi<Dirac::K,2>({"00"}) >> Hadamard) -> (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|11> + (-4.32978e- 17+i*0.707107)*|01> CNOT21 -> (1+i*0)|00><00| + (1+i*0)|01><11| + (1+i*0)|10><10| + (1+i*0)|11><01| (Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21) -> (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e- 17+i*0.707107)*|11> Bob -> (1+i*0)|00><00| + (1+i*0)|01><01| + (1+i*0)|10><10| + (1+i*0)|11><11| Alice -> (1+i*0)|00><00| + (1+i*0)|01><01| + (1+i*0)|10><10| + (1+i*0)|11><11| Measurements: psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (-6.12323e- 17+i*1)*|11>, Alice : (-6.12323e-17+i*1)*|11>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (-6.12323e- 17+i*1)*|11>, Alice : (-6.12323e-17+i*1)*|11>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (-6.12323e- 17+i*1)*|11>, Alice : (-6.12323e-17+i*1)*|11>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>}
  • 2. psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (-6.12323e- 17+i*1)*|11>, Alice : (-6.12323e-17+i*1)*|11>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} psi = (4.32978e-17+i*0.707107)*|00> + (0+i*0)*|10> + (0+i*0)*|01> + (-4.32978e-17+i*0.707107)*|11>, Bob : (6.12323e- 17+i*1)*|00>, Alice : (6.12323e-17+i*1)*|00>} (10+20*i)*K{"1010"} -> (10+i*20)*|1010> (10+20*i)*B{"1010"} -> <1010|*(10+i*20) (e^(i*pi/4))*B{"1010"}*K{"1010"}*~(e^(i*pi/4)) -> 1+i*0 (K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"})) -> (1+i*0)|1010><1010| (K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"}))*(K{"1010"}*~(e^(i*pi/4))) -> (0.707107-i*0.707107)*|1010> (B{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"})) -> <1010|*(0.707107-i*0.707107) (K{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))) -> (2.22045e-16-i*1)*|10101010> (B{"1010"}*~(e^(i*pi/4)))*(B{"1010"}*~(e^(i*pi/4))) -> <10101010|*(2.22045e-16-i*1) Bra<2>{"10"}*Ket<3>{"100"} -> (1+i*0)*|0> Bra<2>{"11"}*Ket<3>{"100"} -> (0+i*0)*|0> Bra<5>{"10011"}*Ket<3>{"100"} -> <11|*(1+i*0) (Bra<1>{"1"}*Bra<1>{"0"})*(Ket<1>{"1"}*Ket<1>{"0"}*Ket<1>{"0"}) -> (1+i*0)*|0> (Bra<1>{"1"}*Bra<1>{"1"})*(Ket<1>{"1"}*Ket<2>{"00"}) -> (0+i*0)*|0> Bra<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*(Bra<2>{"10"}*Bra<1>{"0"}) -> (1+i*0)|00><10011100| Ket<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*(Bra<2>{"10"}*Bra<1>{"0"}) -> (1+i*0)|1100><10011100| std::cos(pi/4)*Ket<1>{"0"} + std::sin(pi/4)*(e^(i*pi/4))*Ket<1>{"1"} ->(0.5+i*0.5)*|1> + (0.707107+i*0)*|0> ((elem*Ket<1>{"0"}*Bra<1>{"0"} - elem*Ket<1>{"0"}*Bra<1>{"1"} + (elem*Ket<1>{"1"}*Bra<1>{"0"} + elem*Ket<1>{"1"}*Bra<1>{"1"}))) -> (0.707107+i*0)|0><0| + (-0.707107+i*0)|0><1| + (0.707107+i*0)|1><0| + (0.707107+i*0)|1><1| 10*i -> +i*10, i*10 -> +i*10 (10*(-i))*(10*i) -> 100 abs(10*i) -> 10, abs(i*10) -> 10 10 +100*i -> 10+i*100, 2*(10 +100*i) -> 20+i*200, (10 +100*i)*3 -> 30+i*300, 4*C{10,100*i} -> 40+i*400, C{10,100*i}*5 -> 50+i*500 (2+3*i)*C{4,5*i}-> -7+i*22 ~(2+3*i)*C{2,3*i} -> 13+i*0 abs(~(2+3*i)*C{2,3*i}) -> 13 (e^(i*pi/4)).m_real -> 0.707107, abs((e^(i*pi/4)).m_real) -> 0.707107 (1000.0*e^(-i*pi/4)) -> 707.107-i*707.107, 1000*~(e^(i*pi/4)) -> 707.107-i*707.107 (e^(i*pi/4))*(e^(i*pi/4)) -> 2.22045e-16+i*1 ~(e^(i*pi/4))*(e^(i*pi/4)) -> 1+i*0 Code: int main(void) { using namespace QuantumGates; typedef QuantumGates::Ket<4> K; typedef QuantumGates::Bra<4> B; std::cout << "===========================================" << std::endl; std::cout << "Usage model:" << std::endl; std::cout << "===========================================" << std::endl; std::cout << "e^(i*theta): Polar notation cos theta + i sin theta" << std::endl; std::cout << "a + i*b: Cartesian notation a + i b" << std::endl; std::cout << "(e^(i*p/4))*Bra<3>{"010"}: Creates 3-qubit bra e^(i pi/4) " "<010|" << std::endl; std::cout << "Ket<3>{"010"}: Creates 3-qubit ket |010>" << std::endl; std::cout << "Bra<3>{"010"}*Ket<3>{"010"}: Inner product <010|010> = 1" << std::endl; std::cout << "Bra<3>{"010"}*Ket<3>{"110"}: Inner product <010|110> = 0" << std::endl; std::cout << "Bra<2>{"01"}*Ket<3>{"011"}: Inner product <01|011> = " "|1>" << std::endl; std::cout << "Bra<2>{"01"}*Bra<3>{"011"}: tensor prod <01|<011| " "= <01011|" << std::endl; std::cout << "Ket<2>{"01"}*Ket<3>{"011"}: tensor prod |01>|011> = " "|01011>" << std::endl; std::cout << "cos(theta)*Ket<1>{"0"}*Bra<1>{"0"} " "- sin(theta)*Ket<1>{"0"}*Bra<1>{"1"}" "+ sin(theta)*Ket<1>{"1"}*Bra<1>{"0"}" "+ cos(theta)*Ket<1>{"1"}*Bra<1>{"1"}: " << std::endl << " Ry(2*theta) = " << std::endl << " -- --" << std::endl<< " | cos theta |0><0| -sin theta |0><1| |"<< std::endl<< " | sin theta |1><0| cos theta |1><1| |"<< std::endl<< " -- --" << std::endl<<
  • 3. " Passive rotation operator. Applied to" << std::endl << " alpha*Ket<1>{"0"} + beta*Ket<1>{"1"} " " = alpha |0> + beta |1> =" << std::endl<< " -- --" << std::endl<< " | alpha |" << std::endl<< " | beta |" << std::endl<< " -- --" << std::endl; std::cout << "Ket<1>{"0"}*Bra<1>{"0"} & Ket<1>{"1"}*Bra<1>{"1"} -" " operator tensor product |0><0| & |1><1| = |01><01|" << std::endl; std::cout << "(Ket<2>{"00"} >> Hadamard >> CNOT21) - the quantum circuit:" " |00> -- H -- CNOT21 = 1/sqrt_2(|00> + |11>), i.e. entangled" << std::endl; std::cout << "(psi >> M >> bob) - performs a measurement on psi " "using the operator 'bob' " << std::endl; std::cout << "===========================================" << std::endl; std::cout << "End of Usage model:" << std::endl; std::cout << "===========================================" << std::endl; auto gates = Gate::instance(); std::cout << "gates.X -> " << gates.X << std::endl; std::cout << "gates.Y -> " << gates.Y << std::endl; std::cout << "gates.Z -> " << gates.Z << std::endl; std::cout << "gates.H -> " << gates.H << std::endl; std::cout << "gates.S -> " << gates.S << std::endl; std::cout << "gates.T -> " << gates.T << std::endl; auto CNOT = ((Ket<1>{"0"}*Bra<1>{"0"}&gates.I) + (Ket<1>{"1"}*Bra<1>{"1"}&gates.X)); std::cout << "CNOT = ((Ket<1>{"0"}*Bra<1>{"0"}&gates.I) + " "(Ket<1>{"1"}*Bra<1>{"1"}&gates.X)) -> " << CNOT << std::endl; std::cout << "gates.controlled(gates.X) -> " << gates.controlled(gates.X) << std::endl; std::cout << "~((e^(i*pi/4))*Ket<1>{"0"}*Bra<1>{"1"}) " "= ~(e^ip/4 |0><1|) -> " << ~((e^(i*pi/4))*Ket<1>{"0"}*Bra<1>{1}) << std::endl; std::cout << "~gates.S = ~(" << gates.S << ") -> " << ~gates.S << std::endl; std::cout << "Psi<Dirac::K,1>({"0"}) = |0> -> " << Psi<Dirac::K,1>({"0"}) << std::endl; std::cout << "(Psi<Dirac::K,1>({"0"})>>gates.H) - single qubit quantum circuit" " |0> -- H -> " << (Psi<Dirac::K,1>({"0"})>>gates.H) << std::endl; auto Hadamard = ((Ket<1>{"0"}*Bra<1>{"0"}&gates.H) + (Ket<1>{"1"}*Bra<1>{"1"}&gates.I)); std::cout << "(Psi<Dirac::K,2>({"00"}) >> Hadamard) -> " << (Psi<Dirac::K,2>({"00"}) >> Hadamard) << std::endl; auto CNOT21 = Ket<2>{"00"}*Bra<2>{"00"} + Ket<2>{"01"}*Bra<2>{"11"} + Ket<2>{"10"}*Bra<2>{"10"} + Ket<2>{"11"}*Bra<2>{"01"}; std::cout << "CNOT21 -> "<< CNOT21 << std::endl; std::cout << "(Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21) -> " << (Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21) << std::endl; auto psi = (Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21); auto bob = ((Ket<1>{"0"}*Bra<1>{"0"} + Ket<1>{"1"}*Bra<1>{"1"}) & gates.I); auto alice = (gates.I &(Ket<1>{"0"}*Bra<1>{"0"} + Ket<1>{"1"}*Bra<1>{"1"})); std::cout << "Bob -> " << bob << std::endl; std::cout << "Alice -> " << alice << std::endl; std::cout << "Measurements:" << std::endl; for(unsigned i=0; i<10; ++i) { auto psi = (Psi<Dirac::K,2>({"00"}) >> Hadamard >> CNOT21); std::cout << "psi = " << psi << ", "; std::cout << "Bob : " << (psi >> M >> bob) << ", "; std::cout << "Alice : " << (psi >> M >> alice) << "} "; std::cout << std::endl; } std::cout << "(10+20*i)*K{"1010"} -> " << (10+20*i)*K{"1010"}<< std::endl; std::cout << "(10+20*i)*B{"1010"} -> " << (10+20*i)*B{"1010"}<< std::endl; std::cout << "(e^(i*pi/4))*B{"1010"}*K{"1010"}*~(e^(i*pi/4)) -> " << (e^(i*pi/4))*B{"1010"}*K{"1010"}*~(e^(i*pi/4)) << std::endl; std::cout << "(K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"})) -> " <<(K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"})) << std::endl; std::cout << "(K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"}))*"
  • 4. "(K{"1010"}*~(e^(i*pi/4)))" " -> " << (K{"1010"}*~(e^(i*pi/4))*((e^(i*pi/4))*B{"1010"}))* (K{"1010"}*~(e^(i*pi/4))) << std::endl; std::cout << "(B{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))*" "((e^(i*pi/4))*B{"1010"})) -> " << (B{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))* ((e^(i*pi/4))*B{"1010"})) << std::endl; std::cout << "(K{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))) -> " << (K{"1010"}*~(e^(i*pi/4)))*(K{"1010"}*~(e^(i*pi/4))) << std::endl; std::cout << "(B{"1010"}*~(e^(i*pi/4)))*(B{"1010"}*~(e^(i*pi/4))) -> " << (B{"1010"}*~(e^(i*pi/4)))*(B{"1010"}*~(e^(i*pi/4))) << std::endl; std::cout << "Bra<2>{"10"}*Ket<3>{"100"} -> " << Bra<2>{"10"}*Ket<3>{"100"} << std::endl; std::cout << "Bra<2>{"11"}*Ket<3>{"100"} -> " << Bra<2>{"11"}*Ket<3>{"100"} << std::endl; std::cout << "Bra<5>{"10011"}*Ket<3>{"100"} -> " << Bra<5>{"10011"}*Ket<3>{"100"} << std::endl; std::cout << "(Bra<1>{"1"}*Bra<1>{"0"})*(Ket<1>{"1"}*Ket<1>{"0"}*Ket<1>{"0"})" " -> " << (Bra<1>{"1"}*Bra<1>{"0"})*(Ket<1>{"1"}*Ket<1>{"0"}*Ket<1>{"0"}) << std::endl; std::cout << "(Bra<1>{"1"}*Bra<1>{"1"})*(Ket<1>{"1"}*Ket<2>{"00"}) " "-> " << (Bra<1>{"1"}*Bra<1>{"1"})*(Ket<1>{"1"}*Ket<2>{"00"}) << std::endl; std::cout << "Bra<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*" "(Bra<2>{"10"}*Bra<1>{"0"}) -> " << Bra<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*(Bra<2>{"10"}*Bra<1>{"0"}) << std::endl; std::cout << "Ket<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})*" "(Bra<2>{"10"}*Bra<1>{"0"}) -> " << Ket<1>{"1"}*(Ket<3>{"100"}*Bra<5>{"10011"})* (Bra<2>{"10"}*Bra<1>{"0"}) << std::endl; std::cout << "std::cos(pi/4)*Ket<1>{"0"} + std::sin(pi/4)*(e^(i*pi/4))*" "Ket<1>{"1"} ->" << std::cos(pi/4)*Ket<1>{"0"} + std::sin(pi/4)*(e^(i*pi/4))*Ket<1>{"1"} << std::endl; auto elem = 1/std::sqrt(2); std::cout <<"((elem*Ket<1>{"0"}*Bra<1>{"0"} " "- elem*Ket<1>{"0"}*Bra<1>{"1"} " "+ (elem*Ket<1>{"1"}*Bra<1>{"0"} " "+ elem*Ket<1>{"1"}*Bra<1>{"1"})))" " -> " << ((elem*Ket<1>{"0"}*Bra<1>{"0"} - elem*Ket<1>{"0"}*Bra<1>{"1"}+ (elem*Ket<1>{"1"}*Bra<1>{"0"} + elem*Ket<1>{"1"}*Bra<1>{"1"}))) << std::endl; std::cout << "10*i -> " << 10*i << ", i*10 -> " << i*10 << std::endl; std::cout << "(10*(-i))*(10*i) -> " << (10*(-i))*(10*i) << std::endl; std::cout << "abs(10*i) -> " << abs(10*i) << ", abs(i*10) -> " << abs(i*10)<< std::endl; std::cout << "10 +100*i -> " << 10 +100*i << ", 2*(10 +100*i) -> "<< 2*(10 +100*i) << ", (10 +100*i)*3 -> " << (10 +100*i)*3 << ", 4*C{10,100*i} -> " << 4*C{10,100*i} << ", C{10,100*i}*5 -> " << C{10,100*i}*5 << std::endl; std::cout << "(2+3*i)*C{4,5*i}-> " << (2+3*i)*C{4,5*i} << std::endl; std::cout << "~(2+3*i)*C{2,3*i} -> " << ~(2+3*i)*C{2,3*i} << std::endl; std::cout << "abs(~(2+3*i)*C{2,3*i}) -> " << abs(~(2+3*i)*C{2,3*i}) << std::endl; auto x = e^(i*pi/4); std::cout << "(e^(i*pi/4)).m_real -> " << x.m_real << ", abs((e^(i*pi/4)).m_real) -> " << abs(x.m_imag)<<std::endl; std::cout << "(1000.0*e^(-i*pi/4)) -> " << (1000.0*e^(-i*pi/4)) << ", 1000*~(e^(i*pi/4)) -> " << 1000*~(e^(i*pi/4)) << std::endl; std::cout << "(e^(i*pi/4))*(e^(i*pi/4)) -> " << x*x << std::endl; std::cout << "~(e^(i*pi/4))*(e^(i*pi/4)) -> "<< ~x*x << std::endl;