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, 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, ket_plus
from proveit.physics.quantum.QPE import QPE1, _U, _ket_u, _phase, _psi_t_ket, _s
from proveit.physics.quantum.circuits import Gate, Input, 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 = Interval(sub_expr2, sub_expr3)
sub_expr5 = MultiQubitElem(element = Gate(operation = QPE1(_U, t), part = sub_expr1), targets = Interval(one, sub_expr3))
sub_expr6 = ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _ket_u, part = sub_expr1), targets = sub_expr4), one, _s)
sub_expr7 = [ExprRange(sub_expr1, Input(state = ket_plus), one, t), ExprRange(sub_expr1, MultiQubitElem(element = Input(state = _ket_u, part = sub_expr1), targets = sub_expr4), one, _s)]
sub_expr8 = [ExprRange(sub_expr1, sub_expr5, one, t), ExprRange(sub_expr1, sub_expr5, sub_expr2, sub_expr3)]
expr = QcircuitEquiv(Qcircuit(vert_expr_array = VertExprArray(sub_expr7, sub_expr8, [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(), sub_expr6])), Qcircuit(vert_expr_array = VertExprArray(sub_expr7, sub_expr8, [ExprRange(sub_expr1, MultiQubitElem(element = Output(state = _psi_t_ket, part = sub_expr1), targets = Interval(one, t)), one, t), sub_expr6])))

# 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{
\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) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert + \rangle} & \multigate{4}{\textrm{QPE}_1\left(U, t\right)} & \multiqout{3}{\lvert \psi_{t} \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle} \\
\qin{\begin{array}{c}:\\ \left(t - 3\right) \times \\:\end{array}} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \qout{\lvert u \rangle}
} \end{array}\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: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 6 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	9, 10, 8
6	Literal
7	ExprTuple	9, 10, 11
8	ExprTuple	12, 18
9	ExprTuple	13, 14
10	ExprTuple	15, 16
11	ExprTuple	17, 18
12	ExprRange	lambda_map: 19 start_index: 20 end_index: 91
13	ExprRange	lambda_map: 21 start_index: 101 end_index: 84
14	ExprRange	lambda_map: 22 start_index: 101 end_index: 85
15	ExprRange	lambda_map: 23 start_index: 101 end_index: 84
16	ExprRange	lambda_map: 23 start_index: 68 end_index: 69
17	ExprRange	lambda_map: 24 start_index: 101 end_index: 84
18	ExprRange	lambda_map: 25 start_index: 101 end_index: 85
19	Lambda	parameter: 114 body: 26
20	Operation	operator: 78 operands: 27
21	Lambda	parameter: 114 body: 28
22	Lambda	parameter: 114 body: 29
23	Lambda	parameter: 114 body: 30
24	Lambda	parameter: 114 body: 31
25	Lambda	parameter: 114 body: 32
26	Operation	operator: 58 operands: 33
27	ExprTuple	34, 101
28	Operation	operator: 52 operands: 35
29	Operation	operator: 39 operands: 36
30	Operation	operator: 39 operands: 37
31	Operation	operator: 39 operands: 38
32	Operation	operator: 39 operands: 40
33	NamedExprs	state: 41
34	Operation	operator: 112 operand: 84
35	NamedExprs	state: 42
36	NamedExprs	element: 43 targets: 49
37	NamedExprs	element: 44 targets: 45
38	NamedExprs	element: 46 targets: 47
39	Literal
40	NamedExprs	element: 48 targets: 49
41	Operation	operator: 88 operands: 50
42	Operation	operator: 97 operand: 64
43	Operation	operator: 52 operands: 59
44	Operation	operator: 53 operands: 54
45	Operation	operator: 60 operands: 55
46	Operation	operator: 58 operands: 56
47	Operation	operator: 60 operands: 57
48	Operation	operator: 58 operands: 59
49	Operation	operator: 60 operands: 61
50	ExprTuple	62, 63
51	ExprTuple	64
52	Literal
53	Literal
54	NamedExprs	operation: 65 part: 114
55	ExprTuple	101, 69
56	NamedExprs	state: 66 part: 114
57	ExprTuple	101, 84
58	Literal
59	NamedExprs	state: 67 part: 114
60	Literal
61	ExprTuple	68, 69
62	Operation	operator: 94 operands: 70
63	Operation	operator: 71 operands: 72
64	Literal
65	Operation	operator: 73 operands: 74
66	Operation	operator: 75 operand: 84
67	Literal
68	Operation	operator: 78 operands: 77
69	Operation	operator: 78 operands: 79
70	ExprTuple	101, 80
71	Literal
72	ExprTuple	81, 82
73	Literal
74	ExprTuple	83, 84
75	Literal
76	ExprTuple	84
77	ExprTuple	84, 101
78	Literal
79	ExprTuple	84, 85
80	Operation	operator: 108 operands: 86
81	Operation	operator: 97 operand: 91
82	Operation	operator: 88 operands: 89
83	Literal
84	Variable
85	Literal
86	ExprTuple	110, 90
87	ExprTuple	91
88	Literal
89	ExprTuple	92, 93
90	Operation	operator: 94 operands: 95
91	Literal
92	Operation	operator: 108 operands: 96
93	Operation	operator: 97 operand: 101
94	Literal
95	ExprTuple	101, 110
96	ExprTuple	99, 100
97	Literal
98	ExprTuple	101
99	Literal
100	Operation	operator: 102 operands: 103
101	Literal
102	Literal
103	ExprTuple	110, 104, 105, 106, 107
104	Literal
105	Literal
106	Operation	operator: 108 operands: 109
107	Literal
108	Literal
109	ExprTuple	110, 111
110	Literal
111	Operation	operator: 112 operand: 114
112	Literal
113	ExprTuple	114
114	Variable

Expression of type QcircuitEquiv¶

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

Expression of type QcircuitEquiv ¶

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