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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, i, 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 = 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_expr2, sub_expr3), Qmult(Qmult(sub_expr1, sub_expr2), sub_expr3)).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 {\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(\left({_{t}}\langle m \rvert \thinspace {\mathrm {FT}}^{\dag}_{t}\right) \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: 9
operands: 5
4Operationoperator: 9
operands: 6
5ExprTuple13, 14, 8
6ExprTuple7, 8
7Operationoperator: 9
operands: 10
8Operationoperator: 11
operand: 15
9Literal
10ExprTuple13, 14
11Literal
12ExprTuple15
13Operationoperator: 16
operands: 17
14Operationoperator: 18
operand: 56
15Lambdaparameter: 48
body: 21
16Literal
17ExprTuple22, 56
18Literal
19ExprTuple56
20ExprTuple48
21Conditionalvalue: 23
condition: 24
22Variable
23Operationoperator: 25
operands: 26
24Operationoperator: 27
operands: 28
25Literal
26ExprTuple29, 30
27Literal
28ExprTuple48, 31
29Operationoperator: 51
operands: 32
30Operationoperator: 33
operands: 34
31Operationoperator: 35
operands: 36
32ExprTuple37, 38
33Literal
34ExprTuple48, 56
35Literal
36ExprTuple39, 40
37Literal
38Operationoperator: 41
operands: 42
39Literal
40Operationoperator: 43
operands: 44
41Literal
42ExprTuple55, 45, 46, 47, 48
43Literal
44ExprTuple49, 50
45Literal
46Literal
47Literal
48Variable
49Operationoperator: 51
operands: 52
50Operationoperator: 53
operand: 57
51Literal
52ExprTuple55, 56
53Literal
54ExprTuple57
55Literal
56Literal
57Literal