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, 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 = Interval(one, sub_expr3)
sub_expr6 = MultiQubitElem(element = Gate(operation = QPE1(_U, t), part = sub_expr1), targets = sub_expr5)
sub_expr7 = MultiQubitElem(element = Output(state = TensorProd(_psi_t_ket, _ket_u), part = sub_expr1), targets = sub_expr5)
sub_expr8 = [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_expr9 = [ExprRange(sub_expr1, sub_expr6, one, t), ExprRange(sub_expr1, sub_expr6, sub_expr2, sub_expr3)]
expr = QcircuitEquiv(Qcircuit(vert_expr_array = VertExprArray(sub_expr8, sub_expr9, [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 = sub_expr4), one, _s)])), Qcircuit(vert_expr_array = VertExprArray(sub_expr8, sub_expr9, [ExprRange(sub_expr1, sub_expr7, one, t), ExprRange(sub_expr1, sub_expr7, sub_expr2, sub_expr3).with_wrapping_at(2,6)]))).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())

\begin{array}{c} \begin{array}{l} \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{4}{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle {\otimes} \lvert u \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 {\otimes} \lvert u \rangle} \\
\qin{\lvert + \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle} \\
\qin{\lvert u \rangle} & \ghost{\textrm{QPE}_1\left(U, t\right)} & \ghostqout{\lvert \psi_{t} \rangle {\otimes} \lvert u \rangle}
} \end{array}\right) \end{array} \end{array}

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.	()	(1)	('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, 13
9	ExprTuple	14, 15
10	ExprTuple	16, 17
11	ExprTuple	18, 19
12	ExprRange	lambda_map: 20 start_index: 21 end_index: 92
13	ExprRange	lambda_map: 22 start_index: 103 end_index: 84
14	ExprRange	lambda_map: 23 start_index: 103 end_index: 95
15	ExprRange	lambda_map: 24 start_index: 103 end_index: 84
16	ExprRange	lambda_map: 25 start_index: 103 end_index: 95
17	ExprRange	lambda_map: 25 start_index: 64 end_index: 67
18	ExprRange	lambda_map: 26 start_index: 103 end_index: 95
19	ExprRange	lambda_map: 26 start_index: 64 end_index: 67
20	Lambda	parameter: 116 body: 27
21	Operation	operator: 76 operands: 28
22	Lambda	parameter: 116 body: 29
23	Lambda	parameter: 116 body: 30
24	Lambda	parameter: 116 body: 31
25	Lambda	parameter: 116 body: 32
26	Lambda	parameter: 116 body: 33
27	Operation	operator: 57 operands: 34
28	ExprTuple	35, 103
29	Operation	operator: 40 operands: 36
30	Operation	operator: 52 operands: 37
31	Operation	operator: 40 operands: 38
32	Operation	operator: 40 operands: 39
33	Operation	operator: 40 operands: 41
34	NamedExprs	state: 42
35	Operation	operator: 114 operand: 95
36	NamedExprs	element: 43 targets: 46
37	NamedExprs	state: 44
38	NamedExprs	element: 45 targets: 46
39	NamedExprs	element: 47 targets: 49
40	Literal
41	NamedExprs	element: 48 targets: 49
42	Operation	operator: 87 operands: 50
43	Operation	operator: 57 operands: 53
44	Operation	operator: 99 operand: 63
45	Operation	operator: 52 operands: 53
46	Operation	operator: 59 operands: 54
47	Operation	operator: 55 operands: 56
48	Operation	operator: 57 operands: 58
49	Operation	operator: 59 operands: 60
50	ExprTuple	61, 62
51	ExprTuple	63
52	Literal
53	NamedExprs	state: 83 part: 116
54	ExprTuple	64, 67
55	Literal
56	NamedExprs	operation: 65 part: 116
57	Literal
58	NamedExprs	state: 66 part: 116
59	Literal
60	ExprTuple	103, 67
61	Operation	operator: 96 operands: 68
62	Operation	operator: 69 operands: 70
63	Literal
64	Operation	operator: 76 operands: 71
65	Operation	operator: 72 operands: 73
66	Operation	operator: 74 operands: 75
67	Operation	operator: 76 operands: 77
68	ExprTuple	103, 78
69	Literal
70	ExprTuple	79, 80
71	ExprTuple	95, 103
72	Literal
73	ExprTuple	81, 95
74	Literal
75	ExprTuple	82, 83
76	Literal
77	ExprTuple	95, 84
78	Operation	operator: 110 operands: 85
79	Operation	operator: 99 operand: 92
80	Operation	operator: 87 operands: 88
81	Literal
82	Operation	operator: 89 operand: 95
83	Literal
84	Literal
85	ExprTuple	112, 91
86	ExprTuple	92
87	Literal
88	ExprTuple	93, 94
89	Literal
90	ExprTuple	95
91	Operation	operator: 96 operands: 97
92	Literal
93	Operation	operator: 110 operands: 98
94	Operation	operator: 99 operand: 103
95	Variable
96	Literal
97	ExprTuple	103, 112
98	ExprTuple	101, 102
99	Literal
100	ExprTuple	103
101	Literal
102	Operation	operator: 104 operands: 105
103	Literal
104	Literal
105	ExprTuple	112, 106, 107, 108, 109
106	Literal
107	Literal
108	Operation	operator: 110 operands: 111
109	Literal
110	Literal
111	ExprTuple	112, 113
112	Literal
113	Operation	operator: 114 operand: 116
114	Literal
115	ExprTuple	116
116	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 ¶