Questions tagged [random-quantum-circuit]

For questions about quantum circuits having a small (polynomial) number of quantum gates; each gate is defined randomly. Random quantum circuits may be implementable shortly, and be used in noisy, intermediate-scale quantum (NISQ) era. Sampling from a random quantum circuit is likely to be classically hard.

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Maximum entanglement entropy of a random circuit

Consider the random quantum circuit below where the gates are randomly taken from SU(4) accordingly with the Haar measure. I am looking to determine an upper bound on the entanglement entropy between ...
Emilio Pezaroglo's user avatar
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Can we train Parametric Quantum Circuits to map any probability distribution to Normal Distribution?

I am trying to find any references to train PQCs to map any probability distribution to a Normal Distribution. Suppose I have MNIST dataset, I want to apply PQC and make the readout distribution to be ...
Kasi Jaswin's user avatar
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Two qubit Pauli expectation value of $\underset{U}{\mathbb{E}}[U^{\otimes 2} (P_1 \otimes P_2)^{\otimes 2} U^{*\otimes 2}]$

I want to find a value for the expression: $$\underset{U}{\mathbb{E}}[U^{\otimes 2} (P_1 \otimes P_2)^{\otimes 2} U^{*\otimes 2}],$$ where $U$ is a two-qubit unitary operator chosen Haar randomly, $...
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Expected trace distance between two types of random ensembles

Consider a Haar random state on $n$ qubits, and denote it by $|\psi\rangle$. Now consider the following state $$|\phi\rangle = \frac{1}{\sqrt{k}} \sum_{i=1}^{k} |\phi_{1, i} \rangle \otimes |\phi_{2, ...
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Implementation of Quantum Random Projection

I want to implement quantum version of Random Projection Dimensionality Reduction Technique on a dataset. Even after using the latest version of Qiskit, it gives the following error: ...
Abhishek Choudhary's user avatar
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Random quantum circuits with expectation values not close to 0 for large number of qubits

I'm studying output expectation values of random quantum circuits. These random circuits are built using random gates from a universal set. I have noticed that the distribution of the output ...
stopper's user avatar
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How do I prove these gate identities?

How do I approach on solving the below two problems?
sriram Usc's user avatar
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How to alter the result of (somewhat) randomly generated circuits?

I create randomly generated circuits by iterating through a list of the gate set (in my case [$CX,SX,RZ,X$]) and adding the gate to the circuit. (In the case of the $CX$ gate we look at the topology ...
Qubii's user avatar
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2 answers
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Induced measure on the set of density matrices defined through the Ginibre ensemble

I am defining a density matrix via $\rho = \frac{X^\dagger X}{\textrm{tr}(X^\dagger X)}$, where $X$ belongs to the Ginibre ensemble. This results in an induced distribution on the set of density ...
Ghost-of-PPPF's user avatar
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Measure on the unitary space and complexity

I'm currently studying various quantum supremacy protocols and i'm struggling to have a clear and well defined view on the rôle of approximating the Haar-measure (through k-designs ...) and the ...
Johan-Luca's user avatar
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59 views

Randomized benchmarking: How do we calculate the reversal element of RB sequence?

In randomized benchmarking the random sequences consist of random Clifford elements, including a computed reversal element, that should return the qubits to the initial state. Let's assume that we ...
Emmanuel's user avatar
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From what distribution is QuTip's rand_herm sampled from?

I am trying to figure out how QuTip samples Hamiltonians using its rand_herm function. It seems to use SciPy's sparse.rand ...
Silly Goose's user avatar
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How to benchmark approximate random unitary sampling

I'm currently studying a specific sampling "quantum advantage" (sorry for the buzzword) protocol wich consist of periodically driving a random Ising chain (https://iopscience.iop.org/article/...
Johan-Luca's user avatar
3 votes
1 answer
321 views

Integral over Haar measure of squared density matrix of Haar random state is proportional to the identity plus swap operator

I am having some trouble understanding why $\int d\psi (| \psi \rangle \langle \psi | )^{\otimes ^2}\propto \ I+$ SWAP , where $|\psi \rangle =U|\psi _0\rangle$ are Haar random states and $d\psi $ is ...
Andrew Dynneson's user avatar
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Generalizing a brick-wall circuit acting on qubits to acting on qudits

Recently, I have been creating a "brick layer" circuit of random unitary gates (each acting on 2 qubits) which acts on a n qubit register using cirq. A typical circuit looks like this. (...
Luke Michie's user avatar
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Independence in state prepared by independently drawn Haar random gates

Consider independently drawn $2 \times 2$ Haar random unitaries $U_1, U_2, \ldots, U_n$ and $$V = U_1 \otimes U_2 \otimes \cdots U_n.$$ Consider the state $\sigma$ given by $$\sigma = V \rho V^{*}, $$ ...
BlackHat18's user avatar
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Quantum circuit simplification using classical computers

Suppose that we have this kind of circuit where the first unitary operator U is used for the state preparation while the Hadamard operator is used of state detection. Let's say we try to run this ...
William's user avatar
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3 votes
1 answer
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Simulation files for noisy quantum circuits

For our study with Yosi Rinott and Tomer Shoham we need the following simple data: Starting with a quantum circuit $C$ on $n$ qubits we need two files: a) A file of probabilities (or amplitudes) for ...
Gil Kalai's user avatar
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Does the invariance of the Haar measure still hold if you use Clifford gates to approximate the Haar random unitaries?

I am not familiar with the Clifford group - I do know that Clifford unitaries can form a unitary 3-design (from this paper) and can be used to approximate Haar random unitaries, but I don't know how ...
Scott's user avatar
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2 answers
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Vanishing expectation value $|\langle Z_1Z_2...Z_N \rangle|$

I'm doing a research involving expectation values of different observables. I've observed that, given a random Quantum Circuit $U$ with $N$ qubits acting on an inital state $|0\rangle$ in such a way ...
stopper's user avatar
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1 answer
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Prove that uniformly random states have moments ${\bf E}_\psi|\langle x|\psi\rangle|^{2t}\sim1/\binom d t$

Im looking for the moments of Haar random states. Is it true that $\textbf{E}_{\psi\sim \text{Haar}}|\langle x| \psi\rangle|^{2t}\sim \frac{1}{\binom{d}{t}}?$ How does one prove this?
postasguest's user avatar
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How to compute projective probabilities of a given measurement outcome using Stim?

I am looking to calculate the projective probabilities for each measurement outcome in a Clifford circuit and then output its corresponding tableau state. I am performing the measurement in the Z ...
user21113's user avatar
1 vote
1 answer
307 views

How to convert Tableau to Circuit in stim

I am trying to simulate random Clifford circuit. I can use stim.Tableau.random(n) to generate it, but don't know how to convert into stim.Circuit() form. It seems current v1.9 does not have to_circuit(...
Yucheng He's user avatar
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215 views

Generating random circuits with only CNOT gates using qiskit.circuit.random.random_circuit

I am trying to generate a random quantum circuit with 5 qubits using only CNOT gates. I have modified the source code of the qiskit.circuit.random.random_circuit function to only include CNOT gates (...
Anonymous's user avatar
2 votes
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102 views

A question on random quantum states and the uniform distribution

Consider an $n$ qubit Haar random quantum state $|\psi\rangle$. Consider a distribution $\mathcal{D}_1$ over $n$ bit strings defined as $$ p_x = |\langle x| \psi \rangle|^{2}, $$ for $x \in \{0, 1\}^{...
Tom Clancy's user avatar
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1 answer
487 views

Clifford circuit approximation to a random Clifford circuit

Given a random Clifford state on $L$ qubits (defined as an infinite depth Clifford circuit acting on the zero state), what depth Clifford circuit is required to approximate this state to a given ...
as2457's user avatar
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5 votes
1 answer
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What's the deal with quantum random number generators?

Classical computers are usually incapable of generating true random numbers as they are based on deterministic algorithms. To overcome this challenge, one can either use pseudo-random number ...
Mauricio's user avatar
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5 votes
1 answer
348 views

Multiplication by a Haar random unitary two times

Consider a Haar random unitary $U$. I am trying to compute the value (or put a bound on) \begin{equation} \mathbb{E}\left[\left|\langle 0^{n} |U^{2} |0^{n}\rangle\right|^{2}\right]. \end{equation} The ...
BlackHat18's user avatar
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Approximating the average of a rational function with respect to the Haar measure

Suppose I have a state $|\psi\rangle = U|0\rangle$ where $U$ is a $d$-dimensional unitary sampled uniformly with respect the Haar measure. I'm interested in computing or approximating analytically an ...
forky40's user avatar
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understanding QuantumCircuit.x() function

For Code 1 qc.x(0,1) throws error but not for code 2 . Please help me to understand code 1 ...
Vinay Sharma's user avatar
2 votes
1 answer
148 views

What is the average amount of gates needed to implement a random Clifford gate?

Given a Clifford gate acting on $n$ qubits is implemented using its generators, what is the average number of gates needed to implement a random Clifford gate as a function of $n$?
Quantum Guy 123's user avatar
1 vote
0 answers
150 views

The phase of the eigenvalue for the LCU method and randomized product formula

If we implement the linear combination of unitary(LCU), $\tilde{U}=\sum_i p_i U_i$, onto its eigenstate, $|\psi \rangle$, we can express its eigenvalue as $re^{i\theta}$, where $\sum_ip_i=1$, $0\leq ...
roaming's user avatar
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6 votes
1 answer
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What is the relationship between the size of the Hilbert space for boson sampling and the complexity of classical simulating it?

My intuition is that the fastest classical algorithm for simulating some kind of noiseless quantum sampling process should scale roughly with the dimension of the Hilbert space: you would need to ...
tparker's user avatar
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2 votes
1 answer
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At what depth and for what architecture are random quantum circuits $1$-designs?

I was confused about something related to quantum $1$ designs. Let us recap two facts we know about random circuit ensembles that form a $1$ design. $1$ design, for a quantum circuit over $n$ qubits, ...
BlackHat18's user avatar
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Random circuits with google cirq

For my dissertation I need to simulate random circuits and I have been trying to use google Cirq for that. Looking at the documentation I have seen how to create my own circuit and simulate it, but it ...
Pablo's user avatar
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2 votes
2 answers
180 views

Random quantum circuits and general efficient POVM measurement

Let's consider a random quantum circuit $C$, applied to the $n$ qubit initial state $|0^{n}\rangle$, producing the state $|\psi\rangle$. Consider a general efficiently implementable $m$-outcome POVM ...
BlackHat18's user avatar
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2 votes
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228 views

Spreading of entanglement with depth for Haar random states

Consider a Haar random quantum state of depth $d$. Consider any bipartition of this state. According to this paper (page $2$): Haar-random states on $n$ qudits are nearly maximally entangled across ...
BlackHat18's user avatar
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4 votes
1 answer
179 views

Spoofing XQUATH with the Feynman method

Consider the XQUATH conjecture for random quantum circuits, as mentioned here. (XQUATH, or Linear Cross-Entropy Quantum Threshold Assumption). There is no polynomial-time classical algorithm that ...
BlackHat18's user avatar
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4 votes
0 answers
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Feynman method and polynomial time algorithm for XQUATH

Consider the Feynman algorithm for simulating quantum circuits, as given here. Consider the XQUATH conjecture for random quantum circuits from here, given by (XQUATH, or Linear Cross-Entropy Quantum ...
BlackHat18's user avatar
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2 votes
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What constitutes generic dynamics, and how is it different from a fully random function?

What constitutes generic dynamics? And how is it different from a fully random function? From what I understand, a fully random function is one that is "Haar" random. And generic dynamics, ...
Ghost-of-PPPF's user avatar
1 vote
1 answer
99 views

What is the counting argument for the number of elementary operations required for a random function?

What is the counting argument for the following statement (classical)? "A random function on n bits requires $e^{\Omega(n)}$ elementary operations." It appears in the introduction of PRL 116,...
Ghost-of-PPPF's user avatar
1 vote
1 answer
399 views

Quantum supremacy: shallow depth Haar random circuits and unitary designs

I had a confusion about shallow depth Haar random quantum circuits. In this paper, in Section B (related works), it is mentioned that Haar random quantum circuits form approximate $2$-designs only ...
BlackHat18's user avatar
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2 votes
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Quantum PRGN against Hard disk Forensics

We can make use of the Qbit to make a random string of bytes then as I have an office laptop and I can do some espionage by copying the files to my local disk from the network share and upload it to ...
Aayush's user avatar
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3 votes
2 answers
701 views

Computing expectation value of $|\langle z|C|0^n\rangle|^2$ over Haar random circuit

I am trying to understand the integration on page 4 of this paper. Consider a Haar random circuit $C$ and a fixed basis $z$. Each output probability of a Haar random circuit (given by $|\langle z | C |...
BlackHat18's user avatar
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1 vote
2 answers
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Why is the Measurement Result Always 1? (expected to find uniformly random measurement)

I created a $|0\rangle$ state then applied $H$ gate to get $\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ and then I meausred my state. But I always found 1. I expected to find 0 and 1 uniformly random ...
quest's user avatar
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4 votes
1 answer
274 views

Compute the large $n$ distribution of $|\langle z_i|\psi\rangle|^2$ over Haar random quantum states

Let $|\psi\rangle$ be a $n$ qubit Haar-random quantum state. I am trying to show that in the limit of large $n$, for each $z_{i} \in \{0, 1\}^{n}$, $$ |\langle 0|\psi\rangle|^{2}, |\langle 1|\psi\...
BlackHat18's user avatar
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7 votes
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Could random quantum circuits be efficiently approximately simulated?

Google's landmark result last year was to compute a task with a quantum computer that a classical computer could not compute, and they chose random circuit sampling. Part of their justification was ...
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What is the Clifford gates selection probability distribution used in the generation of randomized benchmarking circuits?

I've read that in standard randomized benchmarking implementations the random quantum circuits are generated through random gate selection from a uniformly distributed Clifford set of either 1 or 2-...
MShakeG's user avatar
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1 answer
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Implementing a circuit that returns $|01\rangle$ and $|10\rangle$ with equal probability

Using Python how can I implement a quantum circuit that returns $|01\rangle$ or $|10\rangle$ using only $CX$, $RX$ and $RY$ gates, starting with random parametric gates as parameters and optimizing it ...
Rahman Turtle's user avatar
2 votes
1 answer
647 views

How to split a Quantum Circuit on a barrier in Qiskit?

Let's say I have a QuantumCircuit with multiple barriers as shown in the visual below: How would I split up the ...
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