Implement a Hamiltonian in O(n) - exercise question
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I have the following exercise to solve:
Consider the Boolean function $f(x_1 . . . x_n) = x_1 oplus dots oplus x_n$
where $x_1 dots x_n$ is an nbit string and $oplus$ denotes addition mod $2$.
Describe a circuit of $2$-qubit gates on $n + 1$ qubits that implements the
transformation $| x_1 dots x_n rangle | 0 rangle mapsto | x_1 dots x_n rangle | x_1 oplus dots oplus x_nrangle$
By considering a relationship between $f$ and the $n$-qubit Hamiltonian
$Z otimes dots otimes Z$, show that $V = exp(i Z otimes dots otimes Z t)$, for any
fixed $t > 0$, may be implemented on n qubit lines (with possible use
of further ancillary lines) by a circuit of size $O(n)$ of $1$- and
$2$-qubit gates.
My circuit for the first part of the question is just $n$ CNOT gates controlled by the first register and acting on the $1$-qubit register. So far so good. However, for the second part of the question, I don't understand the relation to the Hamiltonian. I understand that I could use $H^{otimes n}$ to convert $X$ into $Z$ gates but still.
Any help appreciated.
algorithm hamiltonian-simulation
edited May 28 at 14:11
Sanchayan Dutta
8,2694 gold badges18 silver badges64 bronze badges
asked May 28 at 8:16
MarslMarsl
877 bronze badges
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add a comment
|
$begingroup$
I have the following exercise to solve:
Consider the Boolean function $f(x_1 . . . x_n) = x_1 oplus dots oplus x_n$
where $x_1 dots x_n$ is an nbit string and $oplus$ denotes addition mod $2$.
Describe a circuit of $2$-qubit gates on $n + 1$ qubits that implements the
transformation $| x_1 dots x_n rangle | 0 rangle mapsto | x_1 dots x_n rangle | x_1 oplus dots oplus x_nrangle$
By considering a relationship between $f$ and the $n$-qubit Hamiltonian
$Z otimes dots otimes Z$, show that $V = exp(i Z otimes dots otimes Z t)$, for any
fixed $t > 0$, may be implemented on n qubit lines (with possible use
of further ancillary lines) by a circuit of size $O(n)$ of $1$- and
$2$-qubit gates.
My circuit for the first part of the question is just $n$ CNOT gates controlled by the first register and acting on the $1$-qubit register. So far so good. However, for the second part of the question, I don't understand the relation to the Hamiltonian. I understand that I could use $H^{otimes n}$ to convert $X$ into $Z$ gates but still.
Any help appreciated.
algorithm hamiltonian-simulation
edited May 28 at 14:11
Sanchayan Dutta
8,2694 gold badges18 silver badges64 bronze badges
asked May 28 at 8:16
MarslMarsl
877 bronze badges
$endgroup$
add a comment
|
$begingroup$
I have the following exercise to solve:
Consider the Boolean function $f(x_1 . . . x_n) = x_1 oplus dots oplus x_n$
where $x_1 dots x_n$ is an nbit string and $oplus$ denotes addition mod $2$.
Describe a circuit of $2$-qubit gates on $n + 1$ qubits that implements the
transformation $| x_1 dots x_n rangle | 0 rangle mapsto | x_1 dots x_n rangle | x_1 oplus dots oplus x_nrangle$
By considering a relationship between $f$ and the $n$-qubit Hamiltonian
$Z otimes dots otimes Z$, show that $V = exp(i Z otimes dots otimes Z t)$, for any
fixed $t > 0$, may be implemented on n qubit lines (with possible use
of further ancillary lines) by a circuit of size $O(n)$ of $1$- and
$2$-qubit gates.
My circuit for the first part of the question is just $n$ CNOT gates controlled by the first register and acting on the $1$-qubit register. So far so good. However, for the second part of the question, I don't understand the relation to the Hamiltonian. I understand that I could use $H^{otimes n}$ to convert $X$ into $Z$ gates but still.
Any help appreciated.
algorithm hamiltonian-simulation
edited May 28 at 14:11
Sanchayan Dutta
8,2694 gold badges18 silver badges64 bronze badges
asked May 28 at 8:16
MarslMarsl
877 bronze badges
$endgroup$
I have the following exercise to solve:
Consider the Boolean function $f(x_1 . . . x_n) = x_1 oplus dots oplus x_n$
where $x_1 dots x_n$ is an nbit string and $oplus$ denotes addition mod $2$.
Describe a circuit of $2$-qubit gates on $n + 1$ qubits that implements the
transformation $| x_1 dots x_n rangle | 0 rangle mapsto | x_1 dots x_n rangle | x_1 oplus dots oplus x_nrangle$
By considering a relationship between $f$ and the $n$-qubit Hamiltonian
$Z otimes dots otimes Z$, show that $V = exp(i Z otimes dots otimes Z t)$, for any
fixed $t > 0$, may be implemented on n qubit lines (with possible use
of further ancillary lines) by a circuit of size $O(n)$ of $1$- and
$2$-qubit gates.
My circuit for the first part of the question is just $n$ CNOT gates controlled by the first register and acting on the $1$-qubit register. So far so good. However, for the second part of the question, I don't understand the relation to the Hamiltonian. I understand that I could use $H^{otimes n}$ to convert $X$ into $Z$ gates but still.
Any help appreciated.
algorithm hamiltonian-simulation
algorithm hamiltonian-simulation
edited May 28 at 14:11
Sanchayan Dutta
8,2694 gold badges18 silver badges64 bronze badges
asked May 28 at 8:16
MarslMarsl
877 bronze badges
edited May 28 at 14:11
Sanchayan Dutta
8,2694 gold badges18 silver badges64 bronze badges
asked May 28 at 8:16
MarslMarsl
877 bronze badges
edited May 28 at 14:11
Sanchayan Dutta
8,2694 gold badges18 silver badges64 bronze badges
edited May 28 at 14:11
Sanchayan Dutta
8,2694 gold badges18 silver badges64 bronze badges
edited May 28 at 14:11
Sanchayan Dutta
8,2694 gold badges18 silver badges64 bronze badges
8,2694 gold badges18 silver badges64 bronze badges
asked May 28 at 8:16
MarslMarsl
877 bronze badges
asked May 28 at 8:16
MarslMarsl
877 bronze badges
asked May 28 at 8:16
MarslMarsl
877 bronze badges
877 bronze badges
add a comment
|
add a comment
|
1 Answer
1
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$begingroup$
Note: I'm deliberately leaving a few gaps here. Hopefully I'm saying enough to let you piece te rest together!
Let's say that you want to implement $V$ on some state
$$
sum_{xin{0,1}^n}alpha_x|xrangle
$$
You can fairly easily write down what that state produces. Think about the Hamiltonian $Z^{otimes n}$. What eigenvalues does it have? $lambda=pm 1$. What are the eigenvectors? The computational basis states $|x_1ldots x_nrangle$. So, the output of $V$ is
$$
sum_{x:lambda_x=1}alpha_xe^{i t}|xrangle+sum_{x:lambda_x=-1}alpha_xe^{-i t}|xrangle
$$
Hence, can you use your function $f$ to determine which eigenvalue a particular $x$ has? How do you then use tha to apply the correct phase for the evolution? Don't forget that if you use an ancilla qubit, you must undo any entanglement you may have created with it.
answered May 28 at 10:13
DaftWullieDaftWullie
21k1 gold badge9 silver badges53 bronze badges
$endgroup$
$begingroup$
perfect explanation, got it thanks, I just use my function to compute whether I have even or odd number of say 1 bits in my state, store this information in an ancilla qubit by the circuit of CX gates, then apply a phase gate to the ancilla, then I undo the first computation by applying the CNOT gates in reverse order (CNOT$^-1=$CNOT) and then I discard the ancilla, but the phase remains on my state?!
$endgroup$
– Marsl
May 29 at 9:52
$begingroup$
@Marsl Yes, exactly :)
$endgroup$
– DaftWullie
May 29 at 10:05
add a comment
|
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1 Answer
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1 Answer
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active
oldest
votes
$begingroup$
Note: I'm deliberately leaving a few gaps here. Hopefully I'm saying enough to let you piece te rest together!
Let's say that you want to implement $V$ on some state
$$
sum_{xin{0,1}^n}alpha_x|xrangle
$$
You can fairly easily write down what that state produces. Think about the Hamiltonian $Z^{otimes n}$. What eigenvalues does it have? $lambda=pm 1$. What are the eigenvectors? The computational basis states $|x_1ldots x_nrangle$. So, the output of $V$ is
$$
sum_{x:lambda_x=1}alpha_xe^{i t}|xrangle+sum_{x:lambda_x=-1}alpha_xe^{-i t}|xrangle
$$
Hence, can you use your function $f$ to determine which eigenvalue a particular $x$ has? How do you then use tha to apply the correct phase for the evolution? Don't forget that if you use an ancilla qubit, you must undo any entanglement you may have created with it.
answered May 28 at 10:13
DaftWullieDaftWullie
21k1 gold badge9 silver badges53 bronze badges
$endgroup$
$begingroup$
perfect explanation, got it thanks, I just use my function to compute whether I have even or odd number of say 1 bits in my state, store this information in an ancilla qubit by the circuit of CX gates, then apply a phase gate to the ancilla, then I undo the first computation by applying the CNOT gates in reverse order (CNOT$^-1=$CNOT) and then I discard the ancilla, but the phase remains on my state?!
$endgroup$
– Marsl
May 29 at 9:52
$begingroup$
@Marsl Yes, exactly :)
$endgroup$
– DaftWullie
May 29 at 10:05
add a comment
|
$begingroup$
Note: I'm deliberately leaving a few gaps here. Hopefully I'm saying enough to let you piece te rest together!
Let's say that you want to implement $V$ on some state
$$
sum_{xin{0,1}^n}alpha_x|xrangle
$$
You can fairly easily write down what that state produces. Think about the Hamiltonian $Z^{otimes n}$. What eigenvalues does it have? $lambda=pm 1$. What are the eigenvectors? The computational basis states $|x_1ldots x_nrangle$. So, the output of $V$ is
$$
sum_{x:lambda_x=1}alpha_xe^{i t}|xrangle+sum_{x:lambda_x=-1}alpha_xe^{-i t}|xrangle
$$
Hence, can you use your function $f$ to determine which eigenvalue a particular $x$ has? How do you then use tha to apply the correct phase for the evolution? Don't forget that if you use an ancilla qubit, you must undo any entanglement you may have created with it.
answered May 28 at 10:13
DaftWullieDaftWullie
21k1 gold badge9 silver badges53 bronze badges
$endgroup$
$begingroup$
perfect explanation, got it thanks, I just use my function to compute whether I have even or odd number of say 1 bits in my state, store this information in an ancilla qubit by the circuit of CX gates, then apply a phase gate to the ancilla, then I undo the first computation by applying the CNOT gates in reverse order (CNOT$^-1=$CNOT) and then I discard the ancilla, but the phase remains on my state?!
$endgroup$
– Marsl
May 29 at 9:52
$begingroup$
@Marsl Yes, exactly :)
$endgroup$
– DaftWullie
May 29 at 10:05
add a comment
|
$begingroup$
Note: I'm deliberately leaving a few gaps here. Hopefully I'm saying enough to let you piece te rest together!
Let's say that you want to implement $V$ on some state
$$
sum_{xin{0,1}^n}alpha_x|xrangle
$$
You can fairly easily write down what that state produces. Think about the Hamiltonian $Z^{otimes n}$. What eigenvalues does it have? $lambda=pm 1$. What are the eigenvectors? The computational basis states $|x_1ldots x_nrangle$. So, the output of $V$ is
$$
sum_{x:lambda_x=1}alpha_xe^{i t}|xrangle+sum_{x:lambda_x=-1}alpha_xe^{-i t}|xrangle
$$
Hence, can you use your function $f$ to determine which eigenvalue a particular $x$ has? How do you then use tha to apply the correct phase for the evolution? Don't forget that if you use an ancilla qubit, you must undo any entanglement you may have created with it.
answered May 28 at 10:13
DaftWullieDaftWullie
21k1 gold badge9 silver badges53 bronze badges
$endgroup$
Note: I'm deliberately leaving a few gaps here. Hopefully I'm saying enough to let you piece te rest together!
Let's say that you want to implement $V$ on some state
$$
sum_{xin{0,1}^n}alpha_x|xrangle
$$
You can fairly easily write down what that state produces. Think about the Hamiltonian $Z^{otimes n}$. What eigenvalues does it have? $lambda=pm 1$. What are the eigenvectors? The computational basis states $|x_1ldots x_nrangle$. So, the output of $V$ is
$$
sum_{x:lambda_x=1}alpha_xe^{i t}|xrangle+sum_{x:lambda_x=-1}alpha_xe^{-i t}|xrangle
$$
Hence, can you use your function $f$ to determine which eigenvalue a particular $x$ has? How do you then use tha to apply the correct phase for the evolution? Don't forget that if you use an ancilla qubit, you must undo any entanglement you may have created with it.
answered May 28 at 10:13
DaftWullieDaftWullie
21k1 gold badge9 silver badges53 bronze badges
answered May 28 at 10:13
DaftWullieDaftWullie
21k1 gold badge9 silver badges53 bronze badges
answered May 28 at 10:13
DaftWullieDaftWullie
21k1 gold badge9 silver badges53 bronze badges
answered May 28 at 10:13
DaftWullieDaftWullie
21k1 gold badge9 silver badges53 bronze badges
21k1 gold badge9 silver badges53 bronze badges
$begingroup$
perfect explanation, got it thanks, I just use my function to compute whether I have even or odd number of say 1 bits in my state, store this information in an ancilla qubit by the circuit of CX gates, then apply a phase gate to the ancilla, then I undo the first computation by applying the CNOT gates in reverse order (CNOT$^-1=$CNOT) and then I discard the ancilla, but the phase remains on my state?!
$endgroup$
– Marsl
May 29 at 9:52
$begingroup$
@Marsl Yes, exactly :)
$endgroup$
– DaftWullie
May 29 at 10:05
add a comment
|
$begingroup$
perfect explanation, got it thanks, I just use my function to compute whether I have even or odd number of say 1 bits in my state, store this information in an ancilla qubit by the circuit of CX gates, then apply a phase gate to the ancilla, then I undo the first computation by applying the CNOT gates in reverse order (CNOT$^-1=$CNOT) and then I discard the ancilla, but the phase remains on my state?!
$endgroup$
– Marsl
May 29 at 9:52
$begingroup$
@Marsl Yes, exactly :)
$endgroup$
– DaftWullie
May 29 at 10:05
$begingroup$
perfect explanation, got it thanks, I just use my function to compute whether I have even or odd number of say 1 bits in my state, store this information in an ancilla qubit by the circuit of CX gates, then apply a phase gate to the ancilla, then I undo the first computation by applying the CNOT gates in reverse order (CNOT$^-1=$CNOT) and then I discard the ancilla, but the phase remains on my state?!
$endgroup$
– Marsl
May 29 at 9:52
$begingroup$
perfect explanation, got it thanks, I just use my function to compute whether I have even or odd number of say 1 bits in my state, store this information in an ancilla qubit by the circuit of CX gates, then apply a phase gate to the ancilla, then I undo the first computation by applying the CNOT gates in reverse order (CNOT$^-1=$CNOT) and then I discard the ancilla, but the phase remains on my state?!
$endgroup$
– Marsl
May 29 at 9:52
$begingroup$
@Marsl Yes, exactly :)
$endgroup$
– DaftWullie
May 29 at 10:05
$begingroup$
@Marsl Yes, exactly :)
$endgroup$
– DaftWullie
May 29 at 10:05
add a comment
|
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