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Anything in between quadratic and exponential speedups?

Question There exist a handful of proven quadratic quantum speedups (some examples include [1-3]) and even a few proven exponential quantum speedups (some examples include [4-6]). But there seems to be a dearth in quantum algorithms with proven…
sheesymcdeezy
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Generalized version of the Hadamard test for $\text{Re} \langle \phi | U | \psi \rangle$

I am wondering if it is possible to generalize the Hadamard test for computing $\text{Re} \langle \phi | U | \psi \rangle$ (different states for left and right operands).
francler
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Why is the orbit of a unitary t design a complex projective t design?

The paper Qubit stabilizer states are complex projective 3-designs states in the final paragraph that "any orbit of a unitary t-design is a complex projective t-design." Using this fact one can take the simple proof that the Clifford group is a…
Ian Gershon Teixeira
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Are there tools I can use to test OpenQASM 3 circuits?

I recently added a to_qasm method to stim. An issue I'm having is how to test that the outputs are correct. I can test OpenQASM 2 outputs by giving the output to qiskit, which can parse OpenQASM 2 and execute the result. But it seems that qiskit…
Craig Gidney
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If a quantum error correcting code can correct every single-qubit $X$ and $Z$ error, can it also correct every single-qubit $Y$ error?

Let $\mathcal{C}$ be a given arbitrary $n$ qubit quantum error correcting code which can correct any single qubit $X$ error and any single qubit $Z$ error, i.e., $\{X_i\}_{i=1}^n$ & $\{Z_i\}_{i=1}^n $. Should this code also be able to correct…
FDGod
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Smallest distance 9 self-dual CSS code?

The level-2 concatenated [[7,1,3]] Steane code, and the 4.8.8 color code are both self-dual [[49,1,9]] codes from the CSS family. Is there a distance 9 self-dual CSS code that has less than 49 qubits?
Balint Pato
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Getting intuition on the state-injection relations for the generalized $\exp(-iP \pi/8)$ $T$-gates (ideally using ZX calculus)

In Litinsky's paper, there are many circuits relations, like the one below. The left handside represents the "rotation" $\exp(-i P \phi)$ with $\phi=\pi/8$ with similar definitions for the orange ($\phi=\pi/4$) and gray box ($\phi=\pi/2$) on the…
Marco Fellous-Asiani
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Conditions for entangling $A$ with $C$ via an interaction on $AB$

I have three qubits in subsystems $A$, $B$, $C$. System $A$ initially contains some state $\rho_A$, and $BC$ contains a bipartite pure state $|\psi\rangle_{BC}$. I apply a unitary operation $U$ acting on $\mathcal{H}_A\otimes \mathcal{H}_B$ and then…
forky40
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Would an ambient-pressure, room-temperature superconductor eliminate the need for a dil-fridge in transmon processors?

Although there are many competing designs for quantum computer architectures, a transmon-based superconducting qubit architecture is well-advanced enough to be "in the lead" across various metrics. But, transmon qubits use a dilution refrigerator…
Mark Spinelli
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Smallest qudit error correcting/detecting codes

Consider encoding a qubit into $n$ qubits. It is well known that the smallest error detecting code is in $n=4$ qubits and the smallest error correcting code is in $n=5$ qubits. Is there a similar result for qudits? If we encode a qudit into $n$…
Eric Kubischta
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Is amplitude estimation optimal?

Amplitude estimation requires $O(1/\epsilon)$ measurements if we want to estimate an amplitude to absolute precision $\epsilon$. Is this optimal? Why can't we do better than this? I'm trying to see if there's an explanation in the literature but I'm…
confusion
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Why is Grover's Algorithm considered to be a Quantum Walk?

I have heard it said that Grover's algorithm is (can be modeled as?) a Quantum Walk. In fact, one reason for their popularity is that QW are used in certain Quantum algorithms. I am trying to understand the connection between this algorithm and a…
Andreas132
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How would HSP with $S_N$ work when the automorphism subgroup is (almost) equal to the symmetric group?

The graph isomorphism problem can be reduced to a case of the hidden subgroup problem, with the group $S_N$ and the function $f \colon \pi \mapsto \pi(G)$ where $G$ is some graph, and $\pi \in S_N$. The hidden subgroup is the group $Aut(G)$; the…
Andrew Baker
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Is every $ ((11,2,5)) $ code equivalent to the $ [[11,1,5]] $ stabilizer code?

Two codes are said to be equivalent if their code spaces are related by a non-entangling gate, i.e., a gate from $U(2)^{\otimes n} \rtimes S_n$, the local unitaries together with permutations. It is proven in Corollary 10 of Quantum Codes of Minimum…
Ian Gershon Teixeira
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Simplify the tensor product of two exponentials

If I have a 2-qubits circuit with a Ry rotation gate acting on each one : My unitary transformation performed on the 2-qubits state is written as : $$e^{-i\theta_{1} \sigma_{y}} \otimes e^{-i\theta_{2} \sigma_{y}}$$ I was wondering if I could…
Duen
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