Expression of type Forall ¶

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 ExprRange, U, Variable, VertExprArray, s, t
from proveit.linear_algebra import MatrixMult, ScalarMult, Unitary
from proveit.logic import Equals, Forall, InSet
from proveit.numbers import Add, Exp, Interval, IntervalCO, 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_s, 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 = Forall(instance_param_or_params = [U], instance_expr = Forall(instance_param_or_params = [var_ket_u], instance_expr = 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)), InSet(phase, IntervalCO(zero, one))]), domain = s_ket_domain, condition = normalized_var_ket_u), domain = Unitary(two_pow_s))

# 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())

\forall_{U \in \textrm{U}\left(2^{s}\right)}~\left[\forall_{\lvert u \rangle \in \mathbb{C}^{2^{s}}~|~\left \|\lvert u \rangle\right \| = 1}~\left[\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), \varphi \in \left[0,1\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)\right]\right]

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

	core type	sub-expressions
0	Operation	operator: 16 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 153 body: 4
3	ExprTuple	153
4	Conditional	value: 5 condition: 6
5	Operation	operator: 16 operand: 9
6	Operation	operator: 50 operands: 8
7	ExprTuple	9
8	ExprTuple	153, 10
9	Lambda	parameter: 142 body: 11
10	Operation	operator: 12 operand: 43
11	Conditional	value: 14 condition: 15
12	Literal
13	ExprTuple	43
14	Operation	operator: 16 operand: 19
15	Operation	operator: 31 operands: 18
16	Literal
17	ExprTuple	19
18	ExprTuple	20, 21
19	Lambda	parameter: 160 body: 23
20	Operation	operator: 50 operands: 24
21	Operation	operator: 48 operands: 25
22	ExprTuple	160
23	Conditional	value: 26 condition: 27
24	ExprTuple	142, 28
25	ExprTuple	29, 155
26	Operation	operator: 48 operands: 30
27	Operation	operator: 31 operands: 32
28	Operation	operator: 33 operands: 34
29	Operation	operator: 35 operand: 142
30	ExprTuple	37, 155
31	Literal
32	ExprTuple	38, 39, 40, 41
33	Literal
34	ExprTuple	42, 43
35	Literal
36	ExprTuple	142
37	Operation	operator: 44 operand: 53
38	Operation	operator: 50 operands: 46
39	Operation	operator: 50 operands: 47
40	Operation	operator: 48 operands: 49
41	Operation	operator: 50 operands: 51
42	Literal
43	Operation	operator: 161 operands: 52
44	Literal
45	ExprTuple	53
46	ExprTuple	160, 54
47	ExprTuple	154, 55
48	Literal
49	ExprTuple	56, 57
50	Literal
51	ExprTuple	160, 58
52	ExprTuple	163, 156
53	Operation	operator: 59 operands: 60
54	Literal
55	Operation	operator: 137 operands: 61
56	Operation	operator: 62 operands: 63
57	Operation	operator: 64 operands: 65
58	Operation	operator: 66 operands: 67
59	Literal
60	ExprTuple	68, 69, 70, 71
61	ExprTuple	74, 72
62	Literal
63	ExprTuple	153, 142
64	Literal
65	ExprTuple	73, 142
66	Literal
67	ExprTuple	74, 155
68	ExprTuple	75, 76
69	ExprTuple	77, 78
70	ExprTuple	79, 80
71	ExprTuple	81, 82
72	Operation	operator: 151 operands: 83
73	Operation	operator: 161 operands: 84
74	Literal
75	ExprRange	lambda_map: 85 start_index: 155 end_index: 164
76	ExprRange	lambda_map: 86 start_index: 155 end_index: 156
77	ExprRange	lambda_map: 87 start_index: 155 end_index: 164
78	ExprRange	lambda_map: 87 start_index: 144 end_index: 145
79	ExprRange	lambda_map: 88 start_index: 155 end_index: 164
80	ExprRange	lambda_map: 89 start_index: 155 end_index: 156
81	ExprRange	lambda_map: 90 start_index: 155 end_index: 164
82	ExprRange	lambda_map: 91 start_index: 155 end_index: 156
83	ExprTuple	159, 92
84	ExprTuple	93, 94
85	Lambda	parameter: 143 body: 95
86	Lambda	parameter: 143 body: 96
87	Lambda	parameter: 143 body: 97
88	Lambda	parameter: 143 body: 98
89	Lambda	parameter: 143 body: 99
90	Lambda	parameter: 143 body: 100
91	Lambda	parameter: 143 body: 102
92	Operation	operator: 103 operand: 155
93	Literal
94	Operation	operator: 157 operands: 105
95	Operation	operator: 129 operands: 106
96	Operation	operator: 113 operands: 107
97	Operation	operator: 113 operands: 108
98	Operation	operator: 109 operands: 110
99	Operation	operator: 130 operands: 111
100	Operation	operator: 113 operands: 112
101	ExprTuple	143
102	Operation	operator: 113 operands: 114
103	Literal
104	ExprTuple	155
105	ExprTuple	163, 115, 116, 160
106	NamedExprs	state: 117
107	NamedExprs	element: 118 targets: 126
108	NamedExprs	element: 119 targets: 120
109	Literal
110	NamedExprs	basis: 121
111	NamedExprs	operation: 122
112	NamedExprs	element: 123 targets: 124
113	Literal
114	NamedExprs	element: 125 targets: 126
115	Literal
116	Literal
117	Operation	operator: 127 operand: 139
118	Operation	operator: 129 operands: 136
119	Operation	operator: 130 operands: 131
120	Operation	operator: 137 operands: 132
121	Literal
122	Literal
123	Operation	operator: 135 operands: 133
124	Operation	operator: 137 operands: 134
125	Operation	operator: 135 operands: 136
126	Operation	operator: 137 operands: 138
127	Literal
128	ExprTuple	139
129	Literal
130	Literal
131	NamedExprs	operation: 140 part: 143
132	ExprTuple	155, 145
133	NamedExprs	state: 141 part: 143
134	ExprTuple	155, 164
135	Literal
136	NamedExprs	state: 142 part: 143
137	Literal
138	ExprTuple	144, 145
139	Literal
140	Operation	operator: 146 operands: 147
141	Operation	operator: 148 operands: 149
142	Variable
143	Variable
144	Operation	operator: 151 operands: 150
145	Operation	operator: 151 operands: 152
146	Literal
147	ExprTuple	153, 164
148	Literal
149	ExprTuple	154, 164
150	ExprTuple	164, 155
151	Literal
152	ExprTuple	164, 156
153	Variable
154	Operation	operator: 157 operands: 158
155	Literal
156	Variable
157	Literal
158	ExprTuple	159, 160
159	Operation	operator: 161 operands: 162
160	Variable
161	Literal
162	ExprTuple	163, 164
163	Literal
164	Variable

Expression of type Forall¶

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

Expression of type Forall ¶

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