358 строки
13 KiB
Plaintext
358 строки
13 KiB
Plaintext
// Copyright (c) Microsoft Corporation. All rights reserved.
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// Licensed under the MIT license.
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//////////////////////////////////////////////////////////////////////
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// This file contains reference solutions to all tasks.
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// The tasks themselves can be found in Tasks.qs file.
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// We recommend that you try to solve the tasks yourself first,
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// but feel free to look up the solution if you get stuck.
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//////////////////////////////////////////////////////////////////////
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namespace Quantum.Kata.RippleCarryAdder {
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open Microsoft.Quantum.Measurement;
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open Microsoft.Quantum.Arrays;
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open Microsoft.Quantum.Intrinsic;
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open Microsoft.Quantum.Canon;
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//////////////////////////////////////////////////////////////////
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// Part I. Simple adder outputting to empty Qubits
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//////////////////////////////////////////////////////////////////
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// Task 1.1. Summation of two bits
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operation LowestBitSum_Reference (a : Qubit, b : Qubit, sum : Qubit) : Unit is Adj {
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CNOT(a, sum);
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CNOT(b, sum);
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}
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// Task 1.2. Carry of two bits
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operation LowestBitCarry_Reference (a : Qubit, b : Qubit, carry : Qubit) : Unit is Adj {
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CCNOT(a, b, carry);
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}
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// Task 1.3. One-bit adder
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operation OneBitAdder_Reference (a : Qubit, b : Qubit, sum : Qubit, carry : Qubit) : Unit is Adj {
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LowestBitSum_Reference(a, b, sum);
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LowestBitCarry_Reference(a, b, carry);
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}
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// Task 1.4. Summation of 3 bits
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operation HighBitSum_Reference (a : Qubit, b : Qubit, carryin : Qubit, sum : Qubit) : Unit is Adj {
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CNOT(a, sum);
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CNOT(b, sum);
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CNOT(carryin, sum);
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}
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// Task 1.5. Carry of 3 bits
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operation HighBitCarry_Reference (a : Qubit, b : Qubit, carryin : Qubit, carryout : Qubit) : Unit is Adj {
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CCNOT(a, b, carryout);
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CCNOT(a, carryin, carryout);
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CCNOT(b, carryin, carryout);
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}
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// Task 1.6. Two-bit adder
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operation TwoBitAdder_Reference (a : Qubit[], b : Qubit[], sum : Qubit[], carry : Qubit) : Unit is Adj {
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use internalCarry = Qubit();
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LowestBitSum_Reference(a[0], b[0], sum[0]);
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LowestBitCarry_Reference(a[0], b[0], internalCarry);
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HighBitSum_Reference(a[1], b[1], internalCarry, sum[1]);
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HighBitCarry_Reference(a[1], b[1], internalCarry, carry);
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// Clean up ancillary qubit
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Adjoint LowestBitCarry_Reference(a[0], b[0], internalCarry);
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}
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// Task 1.7. N-bit adder
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operation ArbitraryAdder_Reference (a : Qubit[], b : Qubit[], sum : Qubit[], carry : Qubit) : Unit is Adj {
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let N = Length(a);
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if N == 1 {
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OneBitAdder_Reference(a[0], b[0], sum[0], carry);
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}
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else {
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use internalCarries = Qubit[N-1];
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within {
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LowestBitCarry_Reference(a[0], b[0], internalCarries[0]);
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for i in 1..N-2 {
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HighBitCarry_Reference(a[i], b[i], internalCarries[i-1], internalCarries[i]);
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}
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}
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apply {
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LowestBitSum_Reference(a[0], b[0], sum[0]);
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for i in 1..N-1 {
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HighBitSum_Reference(a[i], b[i], internalCarries[i-1], sum[i]);
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}
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HighBitCarry_Reference(a[N-1], b[N-1], internalCarries[N-2], carry);
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}
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}
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}
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// A slightly simpler solution - more uniform, but slightly slower, and requires one extra qubit
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operation ArbitraryAdder_Simplified_Reference (a : Qubit[], b : Qubit[], sum : Qubit[], carry : Qubit) : Unit is Adj {
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let N = Length(a);
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use internalCarries = Qubit[N];
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let carries = internalCarries + [carry];
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for i in 0 .. N-1 {
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HighBitSum_Reference(a[i], b[i], carries[i], sum[i]);
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HighBitCarry_Reference(a[i], b[i], carries[i], carries[i+1]);
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}
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// Clean up the ancilla
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for i in N-2 .. -1 .. 0 {
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Adjoint HighBitCarry_Reference(a[i], b[i], carries[i], carries[i+1]);
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}
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}
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// The challenge solution - the sum qubits are used to store the carry bits, and the sum is calculated as they get cleaned up
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operation ArbitraryAdder_Challenge_Reference (a : Qubit[], b : Qubit[], sum : Qubit[], carry : Qubit) : Unit is Adj {
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let N = Length(a);
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// Calculate carry bits
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LowestBitCarry_Reference(a[0], b[0], sum[0]);
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for i in 1 .. N-1 {
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HighBitCarry_Reference(a[i], b[i], sum[i - 1], sum[i]);
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}
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CNOT(sum[N-1], carry);
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// Clean sum qubits and compute sum
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for i in N-1 .. -1 .. 1 {
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Adjoint HighBitCarry_Reference(a[i], b[i], sum[i - 1], sum[i]);
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HighBitSum_Reference(a[i], b[i], sum[i - 1], sum[i]);
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}
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Adjoint LowestBitCarry_Reference(a[0], b[0], sum[0]);
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LowestBitSum_Reference(a[0], b[0], sum[0]);
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}
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//////////////////////////////////////////////////////////////////
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// Part II. Simple in-place adder
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//////////////////////////////////////////////////////////////////
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// Task 2.1. In-place summation of two bits
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operation LowestBitSumInPlace_Reference (a : Qubit, b : Qubit) : Unit is Adj {
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CNOT(a, b);
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}
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// Task 2.2. In-place one-bit adder
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operation OneBitAdderInPlace_Reference (a : Qubit, b : Qubit, carry : Qubit) : Unit is Adj {
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LowestBitCarry_Reference(a, b, carry);
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LowestBitSumInPlace_Reference(a, b);
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}
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// Task 2.3. In-place summation of three bits
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operation HighBitSumInPlace_Reference (a : Qubit, b : Qubit, carryin : Qubit) : Unit is Adj {
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CNOT(a, b);
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CNOT(carryin, b);
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}
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// Task 2.4. In-place two-bit adder
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operation TwoBitAdderInPlace_Reference (a : Qubit[], b : Qubit[], carry : Qubit) : Unit is Adj {
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use internalCarry = Qubit();
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// Set up the carry bits
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LowestBitCarry_Reference(a[0], b[0], internalCarry);
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HighBitCarry_Reference(a[1], b[1], internalCarry, carry);
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// Calculate sums and clean up the ancilla
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HighBitSumInPlace_Reference(a[1], b[1], internalCarry);
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Adjoint LowestBitCarry_Reference(a[0], b[0], internalCarry);
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LowestBitSumInPlace_Reference(a[0], b[0]);
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}
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// Task 2.5. In-place N-bit adder
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operation ArbitraryAdderInPlace_Reference (a : Qubit[], b : Qubit[], carry : Qubit) : Unit is Adj {
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let N = Length(a);
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use internalCarries = Qubit[N];
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// Set up the carry bits
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LowestBitCarry_Reference(a[0], b[0], internalCarries[0]);
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for i in 1 .. N-1 {
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HighBitCarry_Reference(a[i], b[i], internalCarries[i - 1], internalCarries[i]);
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}
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CNOT(internalCarries[N-1], carry);
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// Clean up carry bits and compute sum
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for i in N-1 .. -1 .. 1 {
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Adjoint HighBitCarry_Reference(a[i], b[i], internalCarries[i - 1], internalCarries[i]);
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HighBitSumInPlace_Reference(a[i], b[i], internalCarries[i - 1]);
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}
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Adjoint LowestBitCarry_Reference(a[0], b[0], internalCarries[0]);
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LowestBitSumInPlace_Reference(a[0], b[0]);
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}
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//////////////////////////////////////////////////////////////////
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// Part III*. Improved in-place adder
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//////////////////////////////////////////////////////////////////
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// Task 3.1. Majority gate
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operation Majority_Reference (a : Qubit, b : Qubit, c : Qubit) : Unit is Adj {
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CNOT(a, b);
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CNOT(a, c);
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CCNOT(b, c, a);
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}
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// Task 3.2. UnMajority and Add gate
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operation UnMajorityAdd_Reference (a : Qubit, b : Qubit, c : Qubit) : Unit is Adj {
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CCNOT(b, c, a);
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CNOT(a, c);
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CNOT(c, b);
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}
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// Task 3.3. One-bit majority-UMA adder
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operation OneBitMajUmaAdder_Reference (a : Qubit, b : Qubit, carry : Qubit) : Unit is Adj {
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use tempCarry = Qubit();
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Majority_Reference(a, b, tempCarry);
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CNOT(a, carry);
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UnMajorityAdd_Reference(a, b, tempCarry);
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}
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// Task 3.4. Two-bit majority-UMA adder
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operation TwoBitMajUmaAdder_Reference (a : Qubit[], b : Qubit[], carry : Qubit) : Unit is Adj {
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use tempCarry = Qubit();
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// We only need the extra qubit so we have 3 to pass to the majority gate for the lowest bits
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Majority_Reference(a[0], b[0], tempCarry);
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Majority_Reference(a[1], b[1], a[0]);
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// Save last carry bit
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CNOT(a[1], carry);
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// Restore inputs/ancilla and compute sum
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UnMajorityAdd_Reference(a[1], b[1], a[0]);
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UnMajorityAdd_Reference(a[0], b[0], tempCarry);
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}
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// Task 3.5. N-bit majority-UMA adder
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operation ArbitraryMajUmaAdder_Reference (a : Qubit[], b : Qubit[], carry : Qubit) : Unit is Adj {
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let N = Length(a);
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use tempCarry = Qubit();
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let carries = [tempCarry] + a;
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// Compute carry bits
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for i in 0 .. N-1 {
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Majority_Reference(a[i], b[i], carries[i]);
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}
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// Save last carry bit
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CNOT(carries[N], carry);
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// Restore inputs and ancilla, compute sum
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for i in N-1 .. -1 .. 0 {
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UnMajorityAdd_Reference(a[i], b[i], carries[i]);
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}
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}
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//////////////////////////////////////////////////////////////////
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// Part IV*. In-place subtractor
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//////////////////////////////////////////////////////////////////
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// Task 4.1. N-bit subtractor
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operation Subtractor_Reference (a : Qubit[], b : Qubit[], borrowBit : Qubit) : Unit is Adj {
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// transform b into 2ᴺ - 1 - b
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ApplyToEachA(X, b);
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// compute (2ᴺ - 1 - b) + a = 2ᴺ - 1 - (b - a) using existing adder
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// if this produced a carry, then (2ᴺ - 1 - (b - a)) > 2ᴺ - 1, so (b - a) < 0, and we need a borrow
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// this means we can use the carry qubit from the addition as the borrow qubit
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ArbitraryMajUmaAdder_Reference(a, b, borrowBit);
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// transform 2ᴺ - 1 - (b - a) into b - a
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ApplyToEachA(X, b);
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}
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//////////////////////////////////////////////////////////////////
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// Part V. Addition and subtraction modulo 2ᴺ
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//////////////////////////////////////////////////////////////////
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// Task 5.1. Adder Modulo 2ᴺ
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operation AdderModuloN_Reference (a : Qubit[], b : Qubit[], sum : Qubit[]) : Unit is Adj {
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// Modification of challenge solution of task 1.7 (last carry bit isn't calculated).
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// Sum qubits are used to store the carry bits, and the sum is calculated as they get cleaned up.
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// (It's also possible to use a similar modification of regular solution of task 1.7.)
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let N = Length(a);
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// Calculate carry bits and Store them in Sum
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LowestBitCarry_Reference(a[0], b[0], sum[0]);
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for i in 1 .. N-1 {
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HighBitCarry_Reference(a[i], b[i], sum[i - 1], sum[i]);
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}
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// No need to calculated last (N+1)th carry bit, as addition is modulo 2ᴺ.
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// Clean sum qubits (uncompute carries after they aren't needed) and compute sum
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for i in N-1 .. -1 .. 1 {
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Adjoint HighBitCarry_Reference(a[i], b[i], sum[i - 1], sum[i]); // Restore sum[i] to state 0
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HighBitSum_Reference(a[i], b[i], sum[i - 1], sum[i]); // Compute sum[i]
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}
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Adjoint LowestBitCarry_Reference(a[0], b[0], sum[0]); // Restore sum[0] to state 0
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LowestBitSum_Reference(a[0], b[0], sum[0]); // Compute sum[i]
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}
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// Task 5.2. Two's Complement
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operation TwosComplement_Reference (a : Qubit[]) : Unit is Adj {
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// Transform a into 2ᴺ - 1 - a ("one's complement")
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ApplyToEachA(X, a);
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// Increment mod 2ᴺ.
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// Since we are incrementing by 1, we flip the next most significant bit only if all previous bits are 0.
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for prefix in Prefixes(a) {
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(ControlledOnInt(0, X))(Most(prefix), Tail(prefix));
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}
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}
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// Task 5.3. Subtractor Modulo 2ᴺ
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operation SubtractorModuloN_Reference (a : Qubit[], b : Qubit[], diff : Qubit[]) : Unit is Adj {
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within {
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// Transform a into its Two's Complement 2ᴺ - a
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TwosComplement_Reference(a);
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} apply {
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// Add 2ᴺ - a and b to get (2ᴺ + b - a) mod 2ᴺ = (b-a) mod 2ᴺ
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AdderModuloN_Reference(a, b, diff);
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}
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}
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// Task 5.4. In-place adder modulo 2ᴺ
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operation InPlaceAdderModuloN_Reference (a : Qubit[], b : Qubit[]) : Unit is Adj {
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// Modification of task 3.5 solution (last carry bit isn't calculated)
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use tempCarry = Qubit();
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let carries = [tempCarry] + a;
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// Compute carry bits
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ApplyToEachA(Majority_Reference, Zipped3(a, b, carries));
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// No need to save last (N+1)th carry bit
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// Restore inputs and ancilla, compute sum
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ApplyToEachA(UnMajorityAdd_Reference, Reversed(Zipped3(a, b, carries)));
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}
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// Task 5.5. In-place subtractor modulo 2ᴺ
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operation InPlaceSubtractorModuloN_Reference (a : Qubit[], b : Qubit[]) : Unit is Adj {
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// Notice that Task 5.4 is actually the Adjoint of Task 5.3.
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// Task 5.3 maps (a,b) -> (a,(a+b)mod2ᴺ). Let c = (a+b)mod2ᴺ
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// Task 5.4 maps (a,b) -> (a,(b-a)mod2ᴺ). So Task 5.4 will maps (a,c) -> (a,(c-a)mod2ᴺ) = (a,((a+b)mod2ᴺ-a)mod2ᴺ) = (a,b).
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Adjoint InPlaceAdderModuloN_Reference(a, b);
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}
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}
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