Expression of type QcircuitEquiv

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, Variable, VertExprArray, t
from proveit.linear_algebra import ScalarMult, TensorProd, VecAdd
from proveit.numbers import Add, Exp, Interval, Mult, Neg, e, frac, i, one, pi, sqrt, two, zero
from proveit.physics.quantum import ket0, ket1
from proveit.physics.quantum.QPE import _ket_u, _phase, _psi_t_ket, _s
from proveit.physics.quantum.circuits import MultiQubitElem, Output, Qcircuit, QcircuitEquiv

# 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 = MultiQubitElem(element = Output(state = TensorProd(_psi_t_ket, _ket_u), part = sub_expr1), targets = Interval(one, sub_expr3))
expr = QcircuitEquiv(Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, Output(state = ScalarMult(frac(one, sqrt(two)), VecAdd(ket0, ScalarMult(Exp(e, Mult(two, pi, i, Exp(two, Neg(sub_expr1)), _phase)), ket1)))), Add(Neg(t), one), zero).with_decreasing_order(), ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = Interval(sub_expr2, sub_expr3)), one, _s).with_wrapping_at(2,6)])), Qcircuit(vert_expr_array = VertExprArray([ExprRange(sub_expr1, sub_expr4, one, t), ExprRange(sub_expr1, sub_expr4, sub_expr2, sub_expr3).with_wrapping_at(2,6)])))

expr:

# 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}{c} \Qcircuit@C=1em @R=.7em{
& \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)} \\
& \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)} \\
& \qout{\vdots} \\
& \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)} \\
& \qout{\lvert u \rangle}
} \end{array}\right) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \multiqout{1}{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle} \\
& \ghostqout{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle}
} \end{array}\right)

stored_expr.style_options()

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')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 6 operand: 8
4	Operation	operator: 6 operand: 9
5	ExprTuple	8
6	Literal
7	ExprTuple	9
8	ExprTuple	10, 11
9	ExprTuple	12, 13
10	ExprRange	lambda_map: 14 start_index: 15 end_index: 65
11	ExprRange	lambda_map: 16 start_index: 76 end_index: 57
12	ExprRange	lambda_map: 17 start_index: 76 end_index: 68
13	ExprRange	lambda_map: 17 start_index: 41 end_index: 43
14	Lambda	parameter: 89 body: 18
15	Operation	operator: 50 operands: 19
16	Lambda	parameter: 89 body: 20
17	Lambda	parameter: 89 body: 21
18	Operation	operator: 35 operands: 22
19	ExprTuple	23, 76
20	Operation	operator: 25 operands: 24
21	Operation	operator: 25 operands: 26
22	NamedExprs	state: 27
23	Operation	operator: 87 operand: 68
24	NamedExprs	element: 28 targets: 29
25	Literal
26	NamedExprs	element: 30 targets: 31
27	Operation	operator: 60 operands: 32
28	Operation	operator: 35 operands: 33
29	Operation	operator: 37 operands: 34
30	Operation	operator: 35 operands: 36
31	Operation	operator: 37 operands: 38
32	ExprTuple	39, 40
33	NamedExprs	state: 56 part: 89
34	ExprTuple	41, 43
35	Literal
36	NamedExprs	state: 42 part: 89
37	Literal
38	ExprTuple	76, 43
39	Operation	operator: 69 operands: 44
40	Operation	operator: 45 operands: 46
41	Operation	operator: 50 operands: 47
42	Operation	operator: 48 operands: 49
43	Operation	operator: 50 operands: 51
44	ExprTuple	76, 52
45	Literal
46	ExprTuple	53, 54
47	ExprTuple	68, 76
48	Literal
49	ExprTuple	55, 56
50	Literal
51	ExprTuple	68, 57
52	Operation	operator: 83 operands: 58
53	Operation	operator: 72 operand: 65
54	Operation	operator: 60 operands: 61
55	Operation	operator: 62 operand: 68
56	Literal
57	Literal
58	ExprTuple	85, 64
59	ExprTuple	65
60	Literal
61	ExprTuple	66, 67
62	Literal
63	ExprTuple	68
64	Operation	operator: 69 operands: 70
65	Literal
66	Operation	operator: 83 operands: 71
67	Operation	operator: 72 operand: 76
68	Variable
69	Literal
70	ExprTuple	76, 85
71	ExprTuple	74, 75
72	Literal
73	ExprTuple	76
74	Literal
75	Operation	operator: 77 operands: 78
76	Literal
77	Literal
78	ExprTuple	85, 79, 80, 81, 82
79	Literal
80	Literal
81	Operation	operator: 83 operands: 84
82	Literal
83	Literal
84	ExprTuple	85, 86
85	Literal
86	Operation	operator: 87 operand: 89
87	Literal
88	ExprTuple	89
89	Variable