Expression of type ExprTuple ¶

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, ConditionalSet, ExprRange, ExprTuple, Variable, VertExprArray, t
from proveit.linear_algebra import MatrixExp, ScalarMult, VecAdd
from proveit.logic import Equals, NotEquals, Set
from proveit.numbers import Add, Exp, Interval, Mult, Neg, e, frac, i, one, pi, sqrt, two, zero
from proveit.physics.quantum import CONTROL, ket0, ket1, ket_plus
from proveit.physics.quantum.QPE import QPE1, _U, _ket_u, _phase, _s
from proveit.physics.quantum.circuits import Gate, Igate, Input, MultiQubitElem, Output, Qcircuit, QcircuitEquiv

# build up the expression from sub-expressions
sub_expr1 = Variable("_b", latex_format = r"{_{-}b}")
sub_expr2 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr3 = Add(t, one)
sub_expr4 = Add(sub_expr1, t)
sub_expr5 = Add(t, _s)
sub_expr6 = Interval(sub_expr3, sub_expr5)
sub_expr7 = Add(Neg(t), one)
sub_expr8 = MultiQubitElem(element = Gate(operation = QPE1(_U, t), part = sub_expr2), targets = Interval(one, sub_expr5))
sub_expr9 = [ExprRange(sub_expr2, sub_expr8, one, t), ExprRange(sub_expr2, sub_expr8, sub_expr3, sub_expr5)]
sub_expr10 = [ExprRange(sub_expr2, Input(state = ket_plus), one, t), ExprRange(sub_expr2, MultiQubitElem(element = Input(state = _ket_u, part = sub_expr2), targets = sub_expr6), one, _s)]
sub_expr11 = ExprRange(sub_expr1, [ExprRange(sub_expr2, ConditionalSet(Conditional(MultiQubitElem(element = CONTROL, targets = Set(sub_expr3)), Equals(sub_expr4, sub_expr2)), Conditional(Igate, NotEquals(sub_expr4, sub_expr2))), one, t).with_case_simplification(), ExprRange(sub_expr2, MultiQubitElem(element = Gate(operation = MatrixExp(_U, Exp(two, Neg(sub_expr1))), part = sub_expr2), targets = sub_expr6), one, _s)], sub_expr7, zero).with_decreasing_order()
sub_expr12 = [ExprRange(sub_expr2, Output(state = ScalarMult(frac(one, sqrt(two)), VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr2)), _phase)), ket1)))), sub_expr7, zero).with_decreasing_order(), ExprRange(sub_expr2, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr2), targets = sub_expr6), one, _s)]
expr = ExprTuple(QcircuitEquiv(Qcircuit(vert_expr_array = VertExprArray(sub_expr11)), Qcircuit(vert_expr_array = VertExprArray(sub_expr9))), QcircuitEquiv(Qcircuit(vert_expr_array = VertExprArray(sub_expr10, sub_expr11, sub_expr12)), Qcircuit(vert_expr_array = VertExprArray(sub_expr10, sub_expr9, sub_expr12))).with_wrapping_at(1))

# 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(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \control{} \qw \qwx[1] & \qw & \gate{\cdots} \qwx[1] & \qw & \qw \\
& \qw \qwx[1] & \control{} \qw \qwx[1] & \gate{\cdots} \qwx[1] & \qw & \qw \\
& \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} & \qw \\
& \qw \qwx[1] & \qw \qwx[1] & \gate{\cdots} \qwx[1] & \control{} \qw \qwx[1] & \qw \\
& \gate{U^{2^{t - 1}}} & \gate{U^{2^{t - 2}}} & \gate{\cdots} & \gate{U^{2^{0}}} & { /^{s} } \qw
} \end{array}\right) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \multigate{1}{\textrm{QPE}_1\left(U, t\right)} & { /^{t} } \qw \\
& \ghost{\textrm{QPE}_1\left(U, t\right)} & { /^{s} } \qw
} \end{array}\right), \begin{array}{c} \begin{array}{l} \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \control{} \qw \qwx[1] & \qw & \gate{\cdots} \qwx[1] & \qw & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 1} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
\qin{\lvert + \rangle} & \qw \qwx[1] & \control{} \qw \qwx[1] & \gate{\cdots} \qwx[1] & \qw & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 2} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} & \qout{\vdots} \\
\qin{\lvert + \rangle} & \qw \qwx[1] & \qw \qwx[1] & \gate{\cdots} \qwx[1] & \control{} \qw \qwx[1] & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{0} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
\qin{\lvert u \rangle} & \gate{U^{2^{t - 1}}} & \gate{U^{2^{t - 2}}} & \gate{\cdots} & \gate{U^{2^{0}}} & \qout{\lvert u \rangle}
} \end{array}\right) \\  \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}_1\left(U, t\right)} & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 1} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{t - 2} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \qout{\vdots} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \qout{\frac{1}{\sqrt{2}} \cdot \left(\lvert 0 \rangle + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot 2^{0} \cdot \varphi} \cdot \lvert 1 \rangle\right)\right)} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \qout{\lvert u \rangle}
} \end{array}\right) \end{array} \end{array}\right)

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

name	description	default	current value	related methods
wrap_positions	position(s) at which wrapping is to occur; 'n' is after the nth comma.	()	()	('with_wrapping_at',)
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	left	left	('with_justification',)

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

Expression of type ExprTuple¶

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

Expression of type ExprTuple ¶

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