7 research outputs found
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
Entanglement Scaling in Quantum Advantage Benchmarks
A contemporary technological milestone is to build a quantum device
performing a computational task beyond the capability of any classical
computer, an achievement known as quantum adversarial advantage. In what ways
can the entanglement realized in such a demonstration be quantified? Inspired
by the area law of tensor networks, we derive an upper bound for the minimum
random circuit depth needed to generate the maximal bipartite entanglement
correlations between all problem variables (qubits). This bound is (i) lattice
geometry dependent and (ii) makes explicit a nuance implicit in other proposals
with physical consequence. The hardware itself should be able to support
super-logarithmic ebits of entanglement across some poly() number of
qubit-bipartitions, otherwise the quantum state itself will not possess
volumetric entanglement scaling and full-lattice-range correlations. Hence, as
we present a connection between quantum advantage protocols and quantum
entanglement, the entanglement implicitly generated by such protocols can be
tested separately to further ascertain the validity of any quantum advantage
claim.Comment: updates and improvements from the review process; 8 pages; 3 figure
Quantum Entropy and Central Limit Theorem
We introduce a framework to study discrete-variable (DV) quantum systems
based on qudits. It relies on notions of a mean state (MS), a minimal
stabilizer-projection state (MSPS), and a new convolution. Some interesting
consequences are: The MS is the closest MSPS to a given state with respect to
the relative entropy; the MS is extremal with respect to the von Neumann
entropy, demonstrating a ''maximal entropy principle in DV systems.'' We obtain
a series of inequalities for quantum entropies and for Fisher information based
on convolution, giving a ''second law of thermodynamics for quantum
convolutions.'' We show that the convolution of two stabilizer states is a
stabilizer state. We establish a central limit theorem, based on iterating the
convolution of a zero-mean quantum state, and show this converges to its MS.
The rate of convergence is characterized by the ''magic gap,'' which we define
in terms of the support of the characteristic function of the state. We
elaborate on two examples: the DV beam splitter and the DV amplifier.Comment: 11 pages. See also the companion work arXiv:2302.0842
Magic from a Convolutional Approach
We introduce a convolutional framework to study stabilizer states and
channels based on qudits. This includes the key concepts of a "magic gap," a
mean state (MS), a minimal stabilizer-projection state (MSPS), and a new
convolution. We find that the MS is the closest MSPS to the given state with
respect to relative entropy, and the MS is extremal with respect to von Neumann
entropy. This demonstrates a "maximal entropy principle for DV systems," and
also indicates that the process of taking MS is a nontrivial,
resource-destroying map for magic. We obtain a series of inequalities for
quantum entropies and for Fisher information based on convolution, giving a
"second law of thermodynamics for quantum convolution." The convolution of two
stabilizer states is another stabilizer. We establish a central limit theorem,
based on iterating the convolution of a zero-mean quantum state, and show this
converges to an MS. The rate of convergence is characterized by the magic gap,
which is defined in terms of the support of the characteristic function of the
state. Based on the Choi-Jamiolkowski isomorphism, we introduce the notions of
a mean channel, which is a stabilizer channel, and the convolution of quantum
channels. We obtain results for quantum channels similar to those for states,
and find that Clifford unitaries play an important role in the convolution of
channels in analogous to the role stabilizers play in the convolution of
states. We elaborate these methods with a discussion of three examples: the
qudit DV beam splitter, the qudit DV amplifier and the qubit CNOT gate. All
these results are compatible with the conjecture that stabilizers play the role
in DV quantum systems analogous to Gaussians in continuous-variable quantum
systems.Comment: 46 pages. See also the companion work arXiv:2302.0784