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# 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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### Sampling a Haar Random State Conditioned on Having Low Entanglement Entropy

After applying exponentially, or even polynomially many random local gates to a fixed input state, the resulting distribution of the output state $\lvert\psi_{out}\rangle$ will (be very close) to Haar ...
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1 vote
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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: ...
100 views

### 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 ...
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### How do I prove these gate identities?

How do I approach on solving the below two problems?
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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 ...
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1 vote
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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 ...
1 vote
46 views

### 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 ...
62 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 ...
12 views

### 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 ...
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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/...
453 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 ...
191 views

### 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. (...
111 views

### 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 $$\sigma = V \rho V^{*},$$ ...
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1 vote
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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 ...
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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 ...
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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 ...
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1 vote
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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 ...
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339 views

### 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?
248 views

### 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 ...
1 vote
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### 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(...
263 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 (...
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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 ...
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355 views

### 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, ...
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261 views

### 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 ...
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### 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 ...
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258 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 ...
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### 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 ...
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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 ...
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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, ...
1 vote
105 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,...
1 vote
431 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 ...
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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 ...
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