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Expression of type Equals

from the theory of proveit.physics.quantum.QPE

In [1]:
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 k, m
from proveit.linear_algebra import ScalarMult, VecSum
from proveit.logic import Equals
from proveit.numbers import Exp, Mult, e, frac, i, one, pi, two
from proveit.physics.quantum import NumBra, NumKet, Qmult
from proveit.physics.quantum.QFT import InverseFourierTransform
from proveit.physics.quantum.QPE import _m_domain, _phase, _t
In [2]:
# build up the expression from sub-expressions
sub_expr1 = NumBra(m, _t)
sub_expr2 = InverseFourierTransform(_t)
sub_expr3 = frac(one, Exp(two, frac(_t, two)))
sub_expr4 = VecSum(index_or_indices = [k], summand = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, _t)), domain = _m_domain)
expr = Equals(Qmult(sub_expr1, sub_expr3, sub_expr2, sub_expr4), Qmult(sub_expr3, sub_expr1, sub_expr2, sub_expr4)).with_wrapping_at(2)
expr:
In [3]:
# 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
In [4]:
# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())
\begin{array}{c} \begin{array}{l} \left({_{t}}\langle m \rvert \thinspace \frac{1}{2^{\frac{t}{2}}} \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right)\right) =  \\ \left(\frac{1}{2^{\frac{t}{2}}} \thinspace {_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t} \thinspace \left(\sum_{k=0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \lvert k \rangle_{t}\right)\right)\right) \end{array} \end{array}
In [5]:
stored_expr.style_options()
namedescriptiondefaultcurrent valuerelated methods
operation'infix' or 'function' style formattinginfixinfix
wrap_positionsposition(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.()(2)('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justificationif any wrap positions are set, justify to the 'left', 'center', or 'right'centercenter('with_justification',)
directionDirection of the relation (normal or reversed)normalnormal('with_direction_reversed', 'is_reversed')
In [6]:
# display the expression information
stored_expr.expr_info()
 core typesub-expressionsexpression
0Operationoperator: 1
operands: 2
1Literal
2ExprTuple3, 4
3Operationoperator: 6
operands: 5
4Operationoperator: 6
operands: 7
5ExprTuple9, 8, 10, 11
6Literal
7ExprTuple8, 9, 10, 11
8Operationoperator: 28
operands: 12
9Operationoperator: 13
operands: 14
10Operationoperator: 15
operand: 61
11Operationoperator: 17
operand: 21
12ExprTuple62, 19
13Literal
14ExprTuple20, 61
15Literal
16ExprTuple61
17Literal
18ExprTuple21
19Operationoperator: 56
operands: 22
20Variable
21Lambdaparameter: 53
body: 24
22ExprTuple60, 25
23ExprTuple53
24Conditionalvalue: 26
condition: 27
25Operationoperator: 28
operands: 29
26Operationoperator: 30
operands: 31
27Operationoperator: 32
operands: 33
28Literal
29ExprTuple61, 60
30Literal
31ExprTuple34, 35
32Literal
33ExprTuple53, 36
34Operationoperator: 56
operands: 37
35Operationoperator: 38
operands: 39
36Operationoperator: 40
operands: 41
37ExprTuple42, 43
38Literal
39ExprTuple53, 61
40Literal
41ExprTuple44, 45
42Literal
43Operationoperator: 46
operands: 47
44Literal
45Operationoperator: 48
operands: 49
46Literal
47ExprTuple60, 50, 51, 52, 53
48Literal
49ExprTuple54, 55
50Literal
51Literal
52Literal
53Variable
54Operationoperator: 56
operands: 57
55Operationoperator: 58
operand: 62
56Literal
57ExprTuple60, 61
58Literal
59ExprTuple62
60Literal
61Literal
62Literal