Expression of type Equals ¶

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, InSet
from proveit.numbers import Add, Exp, Interval, IntervalCO, Mult, NaturalPos, Real, e, i, one, pi, subtract, two, zero
from proveit.physics.quantum import I, NumKet, Z, ket_plus, var_ket_u
from proveit.physics.quantum.QPE import QPE, phase, 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))
sub_expr7 = Conditional(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), And(InSet(phase, Real), 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))))
expr = Equals(Conditional(sub_expr7, InSet(t, NaturalPos)), sub_expr7)

# 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\{\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 \textrm{ if } \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)\right.. \textrm{ if } t \in \mathbb{N}^+\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 \textrm{ if } \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)\right..

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	()	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

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

Expression of type Equals¶

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

Expression of type Equals ¶

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