Expression of type Conditional ¶

from the theory of proveit.physics.quantum.QPE ¶

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import Conditional, ExprRange, U, Variable, VertExprArray, s, t
from proveit.linear_algebra import MatrixMult, ScalarMult
from proveit.logic import And, Equals, Forall, InSet
from proveit.numbers import Add, Exp, Interval, Mult, Real, e, i, one, pi, subtract, two, zero
from proveit.physics.quantum import I, NumKet, Z, ket_plus, normalized_var_ket_u, var_ket_u
from proveit.physics.quantum.QPE import QPE, phase, s_ket_domain, two_pow_t
from proveit.physics.quantum.circuits import Gate, Input, Measure, MultiQubitElem, Output, Qcircuit
from proveit.statistics import Prob

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Add(t, one)
sub_expr3 = Add(t, s)
sub_expr4 = Interval(sub_expr2, sub_expr3)
sub_expr5 = Mult(two_pow_t, phase)
sub_expr6 = MultiQubitElem(element = Gate(operation = QPE(U, t), part = sub_expr1), targets = Interval(one, sub_expr3))
expr = Conditional(Forall(instance_param_or_params = [phase], instance_expr = Equals(Prob(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, Input(state = ket_plus), one, t), ExprRange(sub_expr1, MultiQubitElem(element = Input(state = var_ket_u, part = sub_expr1), targets = sub_expr4), one, s)], [ExprRange(sub_expr1, sub_expr6, one, t), ExprRange(sub_expr1, sub_expr6, sub_expr2, sub_expr3)], [ExprRange(sub_expr1, Measure(basis = Z), one, t), ExprRange(sub_expr1, Gate(operation = I).with_implicit_representation(), one, s)], [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = NumKet(sub_expr5, t), part = sub_expr1), targets = Interval(one, t)), one, t), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = var_ket_u, part = sub_expr1), targets = sub_expr4), one, s)]))), one), domain = Real, conditions = [InSet(sub_expr5, Interval(zero, subtract(two_pow_t, one))), Equals(MatrixMult(U, var_ket_u), ScalarMult(Exp(e, Mult(two, pi, i, phase)), var_ket_u))]).with_wrapping(), And(InSet(var_ket_u, s_ket_domain), normalized_var_ket_u))

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left\{\begin{array}{l}\forall_{\varphi \in \mathbb{R}~|~\left(2^{t} \cdot \varphi\right) \in \{0~\ldotp \ldotp~2^{t} - 1\}, \left(U \thinspace \lvert u \rangle\right) = \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi} \cdot \lvert u \rangle\right)}~\\
\left(\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}\left(U, t\right)} & \meter & \multiqout{3}{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}\left(U, t\right)} & \measure{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} \qw & \ghostqout{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & \meter & \ghostqout{\lvert 2^{t} \cdot \varphi \rangle_{t}} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}\left(U, t\right)} & { /^{s} } \qw & \qout{\lvert u \rangle}
} \end{array}\right) = 1\right)\end{array} \textrm{ if } \lvert u \rangle \in \mathbb{C}^{2^{s}} ,  \left \|\lvert u \rangle\right \| = 1\right..

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operand: 6
2	Operation	operator: 18 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	7, 8
6	Lambda	parameter: 142 body: 10
7	Operation	operator: 33 operands: 11
8	Operation	operator: 35 operands: 12
9	ExprTuple	142
10	Conditional	value: 13 condition: 14
11	ExprTuple	124, 15
12	ExprTuple	16, 137
13	Operation	operator: 35 operands: 17
14	Operation	operator: 18 operands: 19
15	Operation	operator: 20 operands: 21
16	Operation	operator: 22 operand: 124
17	ExprTuple	24, 137
18	Literal
19	ExprTuple	25, 26, 27
20	Literal
21	ExprTuple	28, 29
22	Literal
23	ExprTuple	124
24	Operation	operator: 30 operand: 38
25	Operation	operator: 33 operands: 32
26	Operation	operator: 33 operands: 34
27	Operation	operator: 35 operands: 36
28	Literal
29	Operation	operator: 143 operands: 37
30	Literal
31	ExprTuple	38
32	ExprTuple	142, 39
33	Literal
34	ExprTuple	136, 40
35	Literal
36	ExprTuple	41, 42
37	ExprTuple	145, 138
38	Operation	operator: 43 operands: 44
39	Literal
40	Operation	operator: 119 operands: 45
41	Operation	operator: 46 operands: 47
42	Operation	operator: 48 operands: 49
43	Literal
44	ExprTuple	50, 51, 52, 53
45	ExprTuple	54, 55
46	Literal
47	ExprTuple	135, 124
48	Literal
49	ExprTuple	56, 124
50	ExprTuple	57, 58
51	ExprTuple	59, 60
52	ExprTuple	61, 62
53	ExprTuple	63, 64
54	Literal
55	Operation	operator: 133 operands: 65
56	Operation	operator: 143 operands: 66
57	ExprRange	lambda_map: 67 start_index: 137 end_index: 146
58	ExprRange	lambda_map: 68 start_index: 137 end_index: 138
59	ExprRange	lambda_map: 69 start_index: 137 end_index: 146
60	ExprRange	lambda_map: 69 start_index: 126 end_index: 127
61	ExprRange	lambda_map: 70 start_index: 137 end_index: 146
62	ExprRange	lambda_map: 71 start_index: 137 end_index: 138
63	ExprRange	lambda_map: 72 start_index: 137 end_index: 146
64	ExprRange	lambda_map: 73 start_index: 137 end_index: 138
65	ExprTuple	141, 74
66	ExprTuple	75, 76
67	Lambda	parameter: 125 body: 77
68	Lambda	parameter: 125 body: 78
69	Lambda	parameter: 125 body: 79
70	Lambda	parameter: 125 body: 80
71	Lambda	parameter: 125 body: 81
72	Lambda	parameter: 125 body: 82
73	Lambda	parameter: 125 body: 84
74	Operation	operator: 85 operand: 137
75	Literal
76	Operation	operator: 139 operands: 87
77	Operation	operator: 111 operands: 88
78	Operation	operator: 95 operands: 89
79	Operation	operator: 95 operands: 90
80	Operation	operator: 91 operands: 92
81	Operation	operator: 112 operands: 93
82	Operation	operator: 95 operands: 94
83	ExprTuple	125
84	Operation	operator: 95 operands: 96
85	Literal
86	ExprTuple	137
87	ExprTuple	145, 97, 98, 142
88	NamedExprs	state: 99
89	NamedExprs	element: 100 targets: 108
90	NamedExprs	element: 101 targets: 102
91	Literal
92	NamedExprs	basis: 103
93	NamedExprs	operation: 104
94	NamedExprs	element: 105 targets: 106
95	Literal
96	NamedExprs	element: 107 targets: 108
97	Literal
98	Literal
99	Operation	operator: 109 operand: 121
100	Operation	operator: 111 operands: 118
101	Operation	operator: 112 operands: 113
102	Operation	operator: 119 operands: 114
103	Literal
104	Literal
105	Operation	operator: 117 operands: 115
106	Operation	operator: 119 operands: 116
107	Operation	operator: 117 operands: 118
108	Operation	operator: 119 operands: 120
109	Literal
110	ExprTuple	121
111	Literal
112	Literal
113	NamedExprs	operation: 122 part: 125
114	ExprTuple	137, 127
115	NamedExprs	state: 123 part: 125
116	ExprTuple	137, 146
117	Literal
118	NamedExprs	state: 124 part: 125
119	Literal
120	ExprTuple	126, 127
121	Literal
122	Operation	operator: 128 operands: 129
123	Operation	operator: 130 operands: 131
124	Variable
125	Variable
126	Operation	operator: 133 operands: 132
127	Operation	operator: 133 operands: 134
128	Literal
129	ExprTuple	135, 146
130	Literal
131	ExprTuple	136, 146
132	ExprTuple	146, 137
133	Literal
134	ExprTuple	146, 138
135	Variable
136	Operation	operator: 139 operands: 140
137	Literal
138	Variable
139	Literal
140	ExprTuple	141, 142
141	Operation	operator: 143 operands: 144
142	Variable
143	Literal
144	ExprTuple	145, 146
145	Literal
146	Variable

Expression of type Conditional¶

from the theory of proveit.physics.quantum.QPE¶

Expression of type Conditional ¶

from the theory of proveit.physics.quantum.QPE ¶