Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userQuantum homeopathy works: Efficient unitary designs with a system-size independent number of non-Clifford gates
78
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Many quantum information protocols require the implementation of random unitaries. Because it takes exponential resources to produce Haar-random unitaries drawn from the full $n$-qubit group, one often resorts to $t$-designs. Unitary $t$-designs mimic the Haar-measure up to $t$-th moments. It is known that Clifford operations can implement at most $3$-designs. In this work, we quantify the non-Clifford resources required to break this barrier. We find that it suffices to inject $O(t^{4}log^{2}(t)log(1/varepsilon))$ many non-Clifford gates into a polynomial-depth random Clifford circuit to obtain an $varepsilon$-approximate $t$-design. Strikingly, the number of non-Clifford gates required is independent of the system size -- asymptotically, the density of non-Clifford gates is allowed to tend to zero. We also derive novel bounds on the convergence time of random Clifford circuits to the $t$-th moment of the uniform distribution on the Clifford group. Our proofs exploit a recently developed variant of Schur-Weyl duality for the Clifford group, as well as bounds on restricted spectral gaps of averaging operators.
rate research
Read More
We study a method of producing approximately diagonal 1-qubit gates. For each positive integer, the method provides a sequence of gates that are defined iteratively from a fixed diagonal gate and an arbitrary gate. These sequences are conjectured to converge to diagonal gates doubly exponentially fast and are verified for small integers. We systemically study this conjecture and prove several important partial results. Some techniques are developed to pave the way for a final resolution of the conjecture. The sequences provided here have applications in quantum search algorithms, quantum circuit compilation, generation of leakage-free entangled gates in topological quantum computing, etc.
Unitary $t$-designs are `good finite subsets of the unitary group $U(d)$ that approximate the whole unitary group $U(d)$ well. Unitary $t$-designs have been applied in randomized benchmarking, tomography, quantum cryptography and many other areas of quantum information science. If a unitary $t$-design itself is a group then it is called a unitary $t$-group. Although it is known that unitary $t$-designs in $U(d)$ exist for any $t$ and $d$, the unitary $t$-groups do not exist for $tgeq 4$ if $dgeq 3$, as it is shown by Guralnick-Tiep (2005) and Bannai-Navarro-Rizo-Tiep (BNRT, 2018). Explicit constructions of exact unitary $t$-designs in $U(d)$ are not easy in general. In particular, explicit constructions of unitary $4$-designs in $U(4)$ have been an open problem in quantum information theory. We prove that some exact unitary $(t+1)$-designs in the unitary group $U(d)$ are constructed from unitary $t$-groups in $U(d)$ that satisfy certain specific conditions. Based on this result, we specifically construct exact unitary $3$-designs in $U(3)$ from the unitary $2$-group $SL(3,2)$ in $U(3),$ and also unitary $4$-designs in $U(4)$ from the unitary $3$-group $Sp(4,3)$ in $U(4)$ numerically. We also discuss some related problems.
Using a braid group representation based on the Temperley-Lieb algebra, we construct braid quantum gates that could generate entangled $n$-partite $D$-level qudit states. $D$ different sets of $D^ntimes D^n$ unitary representation of the braid group generators are presented. With these generators the desired braid quantum gates are obtained. We show that the generalized GHZ states, which are maximally entangled states, can be obtained directly from these braid quantum gates without resorting to further local unitary transformations. We also point out an interesting observation, namely for a general multi-qudit state there exists a unitary braid quantum gate based on the Temperley-Lieb algebra that connects it from one of its component basis states, if the coefficient of the component state is such that the square of its norm is no less than $1/4$.
We use our Clifford algebra technique, that is nilpotents and projectors which are binomials of the Clifford algebra objects $gamma^a$ with the property ${gamma^a,gamma^b}_+ = 2 eta^{ab}$, for representing quantum gates and quantum algorithms needed in quantum computers in an elegant way. We identify $n$-qubits with spinor representations of the group SO(1,3) for a system of $n$ spinors. Representations are expressed in terms of products of projectors and nilpotents. An algorithm for extracting a particular information out of a general superposition of $2^n$ qubit states is presented. It reproduces for a particular choice of the initial state the Grovers algorithm.
We study the computational power of unitary Clifford circuits with solely magic state inputs (CM circuits), supplemented by classical efficient computation. We show that CM circuits are hard to classically simulate up to multiplicative error (assuming PH non-collapse), and also up to additive error under plausible average-case hardness conjectures. Unlike other such known classes, a broad variety of possible conjectures apply. Along the way we give an extension of the Gottesman-Knill theorem that applies to universal computation, showing that for Clifford circuits with joint stabiliser and non-stabiliser inputs, the stabiliser part can be eliminated in favour of classical simulation, leaving a Clifford circuit on only the non-stabiliser part. Finally we discuss implementational advantages of CM circuits.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
