forked from microsoft/QuantumKatas
-
Notifications
You must be signed in to change notification settings - Fork 0
/
ReferenceImplementation.qs
338 lines (277 loc) · 15.1 KB
/
ReferenceImplementation.qs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
// Copyright (c) Microsoft Corporation. All rights reserved.
// Licensed under the MIT license.
//////////////////////////////////////////////////////////////////////
// This file contains reference solutions to all tasks.
// The tasks themselves can be found in Tasks.qs file.
// We recommend that you try to solve the tasks yourself first,
// but feel free to look up the solution if you get stuck.
//////////////////////////////////////////////////////////////////////
namespace Quantum.Kata.DeutschJozsaAlgorithm {
open Microsoft.Quantum.Arrays;
open Microsoft.Quantum.Diagnostics;
open Microsoft.Quantum.Intrinsic;
open Microsoft.Quantum.Canon;
//////////////////////////////////////////////////////////////////
// Part I. Oracles
//////////////////////////////////////////////////////////////////
// Task 1.1. f(x) = 0
// Inputs:
// 1) N qubits in arbitrary state |x⟩ (input register)
// 2) a qubit in arbitrary state |y⟩ (output qubit)
// Goal: transform state |x, y⟩ into state |x, y ⊕ f(x)⟩ (⊕ is addition modulo 2).
operation Oracle_Zero_Reference (x : Qubit[], y : Qubit) : Unit
is Adj {
// Since f(x) = 0 for all values of x, |y ⊕ f(x)⟩ = |y⟩.
// This means that the operation doesn't need to do any transformation to the inputs.
// Build the project and run the tests to see that T01_Oracle_Zero_Test test passes.
}
// Task 1.2. f(x) = 1
// Inputs:
// 1) N qubits in arbitrary state |x⟩ (input register)
// 2) a qubit in arbitrary state |y⟩ (output qubit)
// Goal: transform state |x, y⟩ into state |x, y ⊕ f(x)⟩ (⊕ is addition modulo 2).
operation Oracle_One_Reference (x : Qubit[], y : Qubit) : Unit
is Adj {
// Since f(x) = 1 for all values of x, |y ⊕ f(x)⟩ = |y ⊕ 1⟩ = |NOT y⟩.
// This means that the operation needs to flip qubit y (i.e. transform |0⟩ to |1⟩ and vice versa).
X(y);
}
// Task 1.3. f(x) = xₖ (the value of k-th qubit)
// Inputs:
// 1) N qubits in arbitrary state |x⟩ (input register)
// 2) a qubit in arbitrary state |y⟩ (output qubit)
// 3) 0-based index of the qubit from input register (0 <= k < N)
// Goal: transform state |x, y⟩ into state |x, y ⊕ xₖ⟩ (⊕ is addition modulo 2).
operation Oracle_Kth_Qubit_Reference (x : Qubit[], y : Qubit, k : Int) : Unit
is Adj {
EqualityFactB(0 <= k and k < Length(x), true, "k should be between 0 and N-1, inclusive");
CNOT(x[k], y);
}
// Task 1.4. f(x) = 1 if x has odd number of 1s, and 0 otherwise
// Inputs:
// 1) N qubits in arbitrary state |x⟩ (input register)
// 2) a qubit in arbitrary state |y⟩ (output qubit)
// Goal: transform state |x, y⟩ into state |x, y ⊕ f(x)⟩ (⊕ is addition modulo 2).
operation Oracle_OddNumberOfOnes_Reference (x : Qubit[], y : Qubit) : Unit
is Adj {
// Hint: f(x) can be represented as x_0 ⊕ x_1 ⊕ ... ⊕ x_(N-1)
for (q in x) {
CNOT(q, y);
}
// alternative solution: ApplyToEachA(CNOT(_, y), x);
}
// Task 1.5. f(x) = Σᵢ 𝑟ᵢ 𝑥ᵢ modulo 2 for a given bit vector r (scalar product function)
// Inputs:
// 1) N qubits in arbitrary state |x⟩ (input register)
// 2) a qubit in arbitrary state |y⟩ (output qubit)
// 3) a bit vector of length N represented as Int[]
// You are guaranteed that the qubit array and the bit vector have the same length.
// Goal: transform state |x, y⟩ into state |x, y ⊕ f(x)⟩ (⊕ is addition modulo 2).
// Note: the functions featured in tasks 1.1, 1.3 and 1.4 are special cases of this function.
operation Oracle_ProductFunction_Reference (x : Qubit[], y : Qubit, r : Int[]) : Unit
is Adj {
// The following line enforces the constraint on the input arrays.
// You don't need to modify it. Feel free to remove it, this won't cause your code to fail.
EqualityFactI(Length(x), Length(r), "Arrays should have the same length");
for (i in IndexRange(x)) {
if (r[i] == 1) {
CNOT(x[i], y);
}
}
}
// Task 1.6. f(x) = Σᵢ (𝑟ᵢ 𝑥ᵢ + (1 - 𝑟ᵢ)(1 - 𝑥ᵢ)) modulo 2 for a given bit vector r
// Inputs:
// 1) N qubits in arbitrary state |x⟩ (input register)
// 2) a qubit in arbitrary state |y⟩ (output qubit)
// 3) a bit vector of length N represented as Int[]
// You are guaranteed that the qubit array and the bit vector have the same length.
// Goal: transform state |x, y⟩ into state |x, y ⊕ f(x)⟩ (⊕ is addition modulo 2).
operation Oracle_ProductWithNegationFunction_Reference (x : Qubit[], y : Qubit, r : Int[]) : Unit
is Adj {
// The following line enforces the constraint on the input arrays.
// You don't need to modify it. Feel free to remove it, this won't cause your code to fail.
EqualityFactI(Length(x), Length(r), "Arrays should have the same length");
for (i in IndexRange(x)) {
if (r[i] == 1) {
CNOT(x[i], y);
} else {
// do a 0-controlled NOT
X(x[i]);
CNOT(x[i], y);
X(x[i]);
}
}
}
// Task 1.7. f(x) = Σᵢ 𝑥ᵢ + (1 if prefix of x is equal to the given bit vector, and 0 otherwise) modulo 2
// Inputs:
// 1) N qubits in arbitrary state |x⟩ (input register)
// 2) a qubit in arbitrary state |y⟩ (output qubit)
// 3) a bit vector of length P represented as Int[] (1 <= P <= N)
// Goal: transform state |x, y⟩ into state |x, y ⊕ f(x)⟩ (⊕ is addition modulo 2).
// A prefix of length k of a state |x⟩ = |x₁, ..., xₙ⟩ is the state of its first k qubits |x₁, ..., xₖ⟩.
// For example, a prefix of length 2 of a state |0110⟩ is 01.
operation Oracle_HammingWithPrefix_Reference (x : Qubit[], y : Qubit, prefix : Int[]) : Unit
is Adj {
// The following line enforces the constraint on the input arrays.
// You don't need to modify it. Feel free to remove it, this won't cause your code to fail.
let P = Length(prefix);
EqualityFactB(1 <= P and P <= Length(x), true, "P should be between 1 and N, inclusive");
// Hint: the first part of the function is the same as in task 1.4
for (q in x) {
CNOT(q, y);
}
// add check for prefix as a multicontrolled NOT
// true bits of r correspond to 1-controls, false bits - to 0-controls
for (i in 0 .. P - 1) {
if (prefix[i] == 0) {
X(x[i]);
}
}
Controlled X(x[0 .. P - 1], y);
// uncompute changes done to input register
for (i in 0 .. P - 1) {
if (prefix[i] == 0) {
X(x[i]);
}
}
}
// Task 1.8*. f(x) = 1 if x has two or three bits (out of three) set to 1, and 0 otherwise (majority function)
// Inputs:
// 1) 3 qubits in arbitrary state |x⟩ (input register)
// 2) a qubit in arbitrary state |y⟩ (output qubit)
// Goal: transform state |x, y⟩ into state |x, y ⊕ f(x)⟩ (⊕ is addition modulo 2).
operation Oracle_MajorityFunction_Reference (x : Qubit[], y : Qubit) : Unit
is Adj {
// The following line enforces the constraint on the input array.
// You don't need to modify it. Feel free to remove it, this won't cause your code to fail.
EqualityFactB(3 == Length(x), true, "x should have exactly 3 qubits");
// f(x) can be represented in terms of AND and ⊕ operations as follows:
// f(x) = (x₀ AND x₁) ⊕ (x₀ AND x₂) ⊕ (x₁ AND x₂)
CCNOT(x[0], x[1], y);
CCNOT(x[0], x[2], y);
CCNOT(x[1], x[2], y);
}
//////////////////////////////////////////////////////////////////
// Part II. Bernstein-Vazirani Algorithm
//////////////////////////////////////////////////////////////////
// Task 2.1. State preparation for Bernstein-Vazirani algorithm
// Inputs:
// 1) N qubits in |0⟩ state (query register)
// 2) a qubit in |0⟩ state (answer register)
// Goal:
// 1) create an equal superposition of all basis vectors from |0...0⟩ to |1...1⟩ on query register
// (i.e. state (|0...0⟩ + ... + |1...1⟩) / sqrt(2^N) )
// 2) create |-⟩ state (|-⟩ = (|0⟩ - |1⟩) / sqrt(2)) on answer register
operation BV_StatePrep_Reference (query : Qubit[], answer : Qubit) : Unit
is Adj {
ApplyToEachA(H, query);
X(answer);
H(answer);
}
// Task 2.2. Bernstein-Vazirani algorithm implementation
// Inputs:
// 1) the number of qubits in the input register N for the function f
// 2) a quantum operation which implements the oracle |x⟩|y⟩ -> |x⟩|y ⊕ f(x)⟩, where
// x is N-qubit input register, y is 1-qubit answer register, and f is a Boolean function
// You are guaranteed that the function f implemented by the oracle is a scalar product function
// (can be represented as f(𝑥₀, …, 𝑥ₙ₋₁) = Σᵢ 𝑟ᵢ 𝑥ᵢ modulo 2 for some bit vector r = (𝑟₀, …, 𝑟ₙ₋₁)).
// You have implemented the oracle implementing the scalar product function in task 1.5.
// Output:
// A bit vector r reconstructed from the function
// Note: a trivial approach is to call the oracle N times:
// |10...0⟩|0⟩ = |10...0⟩|r₀⟩, |010...0⟩|0⟩ = |010...0⟩|r₁⟩ and so on.
// Quantum computing allows to perform this task in just one call to the oracle; try to implement this algorithm.
operation BV_Algorithm_Reference (N : Int, Uf : ((Qubit[], Qubit) => Unit)) : Int[] {
// allocate N qubits for input register and 1 qubit for output
using ((x, y) = (Qubit[N], Qubit())) {
// prepare qubits in the right state
BV_StatePrep_Reference(x, y);
// apply oracle
Uf(x, y);
// apply Hadamard to each qubit of the input register
ApplyToEach(H, x);
// measure all qubits of the input register;
// the result of each measurement is converted to an Int
mutable r = new Int[N];
for (i in 0 .. N - 1) {
if (M(x[i]) != Zero) {
set r w/= i <- 1;
}
}
// before releasing the qubits make sure they are all in |0⟩ state
ResetAll(x);
Reset(y);
return r;
}
}
//////////////////////////////////////////////////////////////////
// Part III. Deutsch-Jozsa Algorithm
//////////////////////////////////////////////////////////////////
// Task 3.1. Deutsch-Jozsa algorithm implementation
// Inputs:
// 1) the number of qubits in the input register N for the function f
// 2) a quantum operation which implements the oracle |x⟩|y⟩ -> |x⟩|y ⊕ f(x)⟩, where
// x is N-qubit input register, y is 1-qubit answer register, and f is a Boolean function
// You are guaranteed that the function f implemented by the oracle is either
// constant (returns 0 on all inputs or 1 on all inputs) or
// balanced (returns 0 on exactly one half of the input domain and 1 on the other half).
// Output:
// true if the function f is constant
// false if the function f is balanced
// Note: a trivial approach is to call the oracle multiple times:
// if the values for more than half of the possible inputs are the same, the function is constant.
// Quantum computing allows to perform this task in just one call to the oracle; try to implement this algorithm.
operation DJ_Algorithm_Reference (N : Int, Uf : ((Qubit[], Qubit) => Unit)) : Bool {
// Declare variable in which the result will be accumulated;
// this variable has to be mutable to allow updating it.
mutable isConstantFunction = true;
// Hint: even though Deutsch-Jozsa algorithm operates on a wider class of functions
// than Bernstein-Vazirani (i.e. functions which can not be represented as a scalar product, such as f(x) = 1),
// it can be expressed as running Bernstein-Vazirani algorithm
// and then post-processing the return value classically:
// the function is constant if and only if all elements of the returned array are false
let r = BV_Algorithm_Reference(N, Uf);
for (i in 0 .. N - 1) {
set isConstantFunction = isConstantFunction and r[i] == 0;
}
return isConstantFunction;
}
//////////////////////////////////////////////////////////////////
// Part IV. Come up with your own algorithm!
//////////////////////////////////////////////////////////////////
// Task 4.1. Reconstruct the oracle from task 1.6
// Inputs:
// 1) the number of qubits in the input register N for the function f
// 2) a quantum operation which implements the oracle |x⟩|y⟩ -> |x⟩|y ⊕ f(x)⟩, where
// x is N-qubit input register, y is 1-qubit answer register, and f is a Boolean function
// You are guaranteed that the function f implemented by the oracle can be represented as
// f(𝑥₀, …, 𝑥ₙ₋₁) = Σᵢ (𝑟ᵢ 𝑥ᵢ + (1 - 𝑟ᵢ)(1 - 𝑥ᵢ)) modulo 2 for some bit vector r = (𝑟₀, …, 𝑟ₙ₋₁).
// You have implemented the oracle implementing this function in task 1.6.
// Output:
// A bit vector r which generates the same oracle as the one you are given
operation Noname_Algorithm_Reference (N : Int, Uf : ((Qubit[], Qubit) => Unit)) : Int[] {
using ((x, y) = (Qubit[N], Qubit())) {
// apply oracle to qubits in all 0 state
Uf(x, y);
// f(x) = Σᵢ (𝑟ᵢ 𝑥ᵢ + (1 - 𝑟ᵢ)(1 - 𝑥ᵢ)) = 2 Σᵢ 𝑟ᵢ 𝑥ᵢ + Σᵢ 𝑟ᵢ + Σᵢ 𝑥ᵢ + N = Σᵢ 𝑟ᵢ + N
// remove the N from the expression
if (N % 2 == 1) {
X(y);
}
// now y = Σᵢ 𝑟ᵢ
// Declare an Int array in which the result will be stored;
// the variable has to be mutable to allow updating it.
mutable r = new Int[N];
// measure the output register
let m = M(y);
if (m == One) {
// adjust parity of bit vector r
set r w/= 0 <- 1;
}
// before releasing the qubits make sure they are all in |0⟩ state
ResetAll(x);
Reset(y);
return r;
}
}
}