28 research outputs found
Classical simulation of quantum circuits by half Gauss sums
We give an efficient algorithm to evaluate a certain class of exponential
sums, namely the periodic, quadratic, multivariate half Gauss sums. We show
that these exponential sums become -hard to compute when we omit
either the periodic or quadratic condition. We apply our results about these
exponential sums to the classical simulation of quantum circuits and give an
alternative proof of the Gottesman-Knill theorem. We also explore a connection
between these exponential sums and the Holant framework. In particular, we
generalize the existing definition of affine signatures to arbitrary dimensions
and use our results about half Gauss sums to show that the Holant problem for
the set of affine signatures is tractable.Comment: 25 pages, no figure
Error-mitigated fermionic classical shadows on noisy quantum devices
Efficiently estimating fermionic Hamiltonian expectation values is vital for simulating various physical systems. Classical shadow (CS) algorithms offer a solution by reducing the number of quantum state copies needed, but noise in quantum devices poses challenges. We propose an error-mitigated CS algorithm assuming gate-independent, time-stationary, and Markovian (GTM) noise. For n-qubit systems, our algorithm, which employs the easily prepared initial state vertical bar 0(n)> < 0(n)vertical bar assumed to be noiseless, efficiently estimates k-RDMs with (O) over tilde (kn(k)) state copies and (O) over tilde(root n) calibration measurements for GTM noise with constant fidelities. We show that our algorithm is robust against noise types like depolarizing, damping, and X-rotation noise with constant strengths, showing scalings akin to prior CS algorithms for fermions but with better noise resilience. Numerical simulations confirm our algorithm's efficacy in noisy settings, suggesting its viability for near-term quantum devices
Pseudorandom unitaries are neither real nor sparse nor noise-robust
Pseudorandom quantum states (PRSs) and pseudorandom unitaries (PRUs) possess
the dual nature of being efficiently constructible while appearing completely
random to any efficient quantum algorithm. In this study, we establish
fundamental bounds on pseudorandomness. We show that PRSs and PRUs exist only
when the probability that an error occurs is negligible, ruling out their
generation on noisy intermediate-scale and early fault-tolerant quantum
computers. Additionally, we derive lower bounds on the imaginarity and
coherence of PRSs and PRUs, rule out the existence of sparse or real PRUs, and
show that PRUs are more difficult to generate than PRSs. Our work also
establishes rigorous bounds on the efficiency of property testing,
demonstrating the exponential complexity in distinguishing real quantum states
from imaginary ones, in contrast to the efficient measurability of unitary
imaginarity. Furthermore, we prove lower bounds on the testing of coherence.
Lastly, we show that the transformation from a complex to a real model of
quantum computation is inefficient, in contrast to the reverse process, which
is efficient. Overall, our results establish fundamental limits on property
testing and provide valuable insights into quantum pseudorandomness.Comment: 23 pages, 3 figure
Complexity Classification of Conjugated Clifford Circuits
Clifford circuits - i.e. circuits composed of only CNOT, Hadamard, and pi/4 phase gates - play a central role in the study of quantum computation. However, their computational power is limited: a well-known result of Gottesman and Knill states that Clifford circuits are efficiently classically simulable. We show that in contrast, "conjugated Clifford circuits" (CCCs) - where one additionally conjugates every qubit by the same one-qubit gate U - can perform hard sampling tasks. In particular, we fully classify the computational power of CCCs by showing that essentially any non-Clifford conjugating unitary U can give rise to sampling tasks which cannot be efficiently classically simulated to constant multiplicative error, unless the polynomial hierarchy collapses. Furthermore, by standard techniques, this hardness result can be extended to allow for the more realistic model of constant additive error, under a plausible complexity-theoretic conjecture. This work can be seen as progress towards classifying the computational power of all restricted quantum gate sets
Doubly infinite separation of quantum information and communication
We prove the existence of (one-way) communication tasks with a subconstant
versus superconstant asymptotic gap, which we call "doubly infinite," between
their quantum information and communication complexities. We do so by studying
the exclusion game [C. Perry et al., Phys. Rev. Lett. 115, 030504 (2015)] for
which there exist instances where the quantum information complexity tends to
zero as the size of the input increases. By showing that the quantum
communication complexity of these games scales at least logarithmically in ,
we obtain our result. We further show that the established lower bounds and
gaps still hold even if we allow a small probability of error. However in this
case, the -qubit quantum message of the zero-error strategy can be
compressed polynomially.Comment: 16 pages, 2 figures. v4: minor errors fixed; close to published
version; v5: financial support info adde