Perimeter Institute Quantum Discussions

We define a map from an arbitrary quantum circuit to a local Hamiltonian whose ground state encodes the quantum computation. All previous maps relied on the Feynman-Kitaev construction, which introduces an ancillary ‘clock register’ to track the computational steps. Our construction, on the other hand, relies on injective tensor networks with associated parent Hamiltonians, avoiding the introduction of a clock register. This comes at the cost of the ground state containing only a noisy version of the quantum computation, with independent stochastic noise. We can remedy this - making our construction robust - by using quantum fault tolerance. In addition to the stochastic noise, we show that any state with energy density exponentially small in the circuit depth encodes a noisy version of the quantum computation with adversarial noise. We also show that any ‘combinatorial state’ with energy density polynomially small in depth encodes the quantum computation with adversarial noise. This serves as evidence that any state with energy density polynomially small in depth has a similar property. As an application, we give a new proof of the QMA-completeness of the local Hamiltonian problem (with logarithmic locality) and show that contracting injective tensor networks to additive error is BQP- hard. We also discuss the implication of our construction to the quantum PCP conjecture, combining with an observation that QMA verification can be done in logarithmic depth. Based on joint work with Anurag Anshu and Nikolas P. Breuckmann. (https://arxiv.org/abs/2309.16475)

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Quantum measurement has conventionally been regarded as the final step in quantum information processing, which is essential for reading out the processed information but collapses the quantum state into a classical state. However, recent studies have shown that quantum measurement itself can induce novel quantum phenomena. One seminal example is a monitored random circuit, which can generate long-range entanglement faster than a random unitary circuit. Inspired by these results, in this talk, we address the following question: When quantum information is encoded in a quantum error-correcting code, how many physical qubits should be randomly measured to destroy the encoded information? We investigate this question for various quantum error-correcting codes and derive the necessary and sufficient conditions for destroying the information through measurements. In particular, we demonstrate that for a large class of quantum error-correcting codes, it is impossible to destroy the encoded information through random single-qubit Pauli measurements when a tiny portion of physical qubits is still unmeasured. Our results not only reveal the extraordinary robustness of quantum codes under measurement decoherence, but also suggest potential applications in quantum information processing tasks.

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Fault-tolerant protocols and quantum error correction (QEC) are essential to building reliable quantum computers from imperfect components that are vulnerable to errors. Optimizing the resource and time overheads needed to implement QEC is one of the most pressing challenges that will facilitate a transition from NISQ to the fault tolerance era. In this talk, I will discuss two intriguing ideas that can significantly reduce these overheads. The first idea, erasure qubits, relies on an efficient conversion of the dominant noise into erasure errors at known locations, greatly enhancing the performance of QEC protocols. The second idea, single-shot QEC, guarantees that even in the presence of measurement errors one can perform reliable QEC without repeating measurements, incurring only constant time overhead.

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With the recent construction of quantum low-density parity-check (LDPC) codes with optimal asymptotic parameters, finding methods to perform low-overhead computation using those constructions has become a central problem of quantum error-correction. In particular, triorthogonal codes---which admit transversal non-Clifford operations---are of particular interest, but few examples of these codes are presently known. In our work, we introduce a new family of codes, the quantum rainbow codes, a generalization of pin codes and color codes, that can be constructed from any chain complex. When applied to the hypergraph product of three complexes, we show that those codes can implement transversal non-Clifford gates and have improved parameters compared to pin codes. Considering expander graphs with large girth as the input complexes, we can for instance obtain families of triorthogonal codes with parameters [[n,Θ(n^{2/3}),Θ(log(n))]].

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Particle physics underpins our understanding of the world at a fundamental level by describing the interplay of matter and forces through gauge theories. Yet, despite their unmatched success, the intrinsic quantum mechanical nature of gauge theories makes important problem classes notoriously difficult to address with classical computational techniques. A promising way to overcome these roadblocks is offered by quantum computers, which are based on the same laws that make the classical computations so difficult. Here, we present a quantum computation of the properties of the basic building block of two-dimensional lattice quantum electrodynamics, involving both gauge fields and matter. This computation is made possible by the use of a trapped-ion qudit quantum processor, where quantum information is encoded in different states per ion, rather than in two states as in qubits. Qudits are ideally suited for describing gauge fields, which are naturally high-dimensional, leading to a dramatic reduction in the quantum register size and circuit complexity. Using a variational quantum eigensolver, we find the ground state of the model and observe the interplay between virtual pair creation and quantized magnetic field effects. The qudit approach further allows us to seamlessly observe the effect of different gauge field truncations by controlling the qudit dimension. Our results open the door for hardware-efficient quantum simulations with qudits in near-term quantum devices.

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The essential requirement for fault-tolerant quantum computation (FTQC) is the total protocol design to achieve a fair balance of all the critical factors relevant to its practical realization, such as the space overhead, the threshold, and the modularity. A major obstacle in realizing FTQC with conventional protocols, such as those based on the surface code and the concatenated Steane code, has been the space overhead, i.e., the required number of physical qubits per logical qubit. Protocols based on high-rate quantum low-density parity-check (LDPC) codes gather considerable attention as a way to reduce the space overhead, but problematically, the existing fault-tolerant protocols for such quantum LDPC codes sacrifice the other factors. Here we construct a new fault-tolerant protocol to meet these requirements simultaneously based on more recent progress on the techniques for concatenated codes rather than quantum LDPC codes, achieving a constant space overhead, a high threshold, and flexibility in modular architecture designs. In particular, under a physical error rate of 0.1%, our protocol reduces the space overhead to achieve the logical CNOT error rates 10^{−10} and 10^{−24} by more than 90% and 97%, respectively, compared to the protocol for the surface code. Furthermore, our protocol achieves the threshold of 2.4% under a conventional circuit-level error model, substantially outperforming that of the surface code. The use of concatenated codes also naturally introduces abstraction layers essential for the modularity of FTQC architectures. These results indicate that the code-concatenation approach opens a way to significantly save qubits in realizing FTQC while fulfilling the other essential requirements for the practical protocol design.

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In this talk, we will outline an efficient algorithm for port-based teleportation, a unitarily equivariant version of teleportation useful for constructing programmable quantum processors and performing instantaneous nonlocal computation (NLQC). The latter connection is important in AdS/CFT, where bulk computations are realized as boundary NLQC. Our algorithm yields an exponential improvement to the known relationship between the amount of entanglement available and the complexity of the nonlocal part of any unitary that can be implemented usin NLQC. Similarly, our algorithm provides the first nontrivial efficient algorithm for an approximate universal programmable quantum processor.

The key to our approach is a general quantum algorithm we develop for block diagonalizing so-called generalized induced representations, a novel type of representation that arises from lifting a representation of a subgroup to one for the whole group while relaxing a linear independence condition from the standard definition. Generalized induced representations appear naturally in quantum information, notably in generalizations of Schur-Weyl duality. For the case of port-based teleportation, we apply this framework to develop an efficient twisted Schur transform for transforming to a subgroup-reduced irrep basis of the partially transposed permutation algebra, whose dual is the U⊗n−k ⊗ (U∗) ⊗k representation of the unitary group.

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The asymptotic theory of quantum channel estimation has been well established, but in general noiseless and controllable ancilla is required for attaining the ultimate limit in the asymptotic regime. Little is known about the metrological performance without noiseless ancilla, which is more relevant in practical circumstances. In this work, we present a novel theoretical framework to address this problem, bridging quantum metrology and the asymptotic theory of quantum channels. Leveraging this framework, we prove sufficient conditions for achieving the Heisenberg limit with repeated application of the channel to estimate, both with and without applying interleaved unitary control operations. For the latter case, we design an algorithm to identify the control operation. Finally, we analyze several intriguing examples by our approach.

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Many quantum platforms naturally host Hamiltonians with power-law or even all-to-all connectivity, which may potentially process quantum information in a way much faster than conventional gate-based models. For such non-geometrically-local Hamiltonians, it is then important to both come up with fast protocols and understand the ultimate limit for realizing various information processing tasks. In this talk, I will first overview this quantum speed limit topic, and then dive into the particular task of quantum routing, i.e. permuting unknown quantum states on the qubits. I aim to show [1] a provably optimal Hamiltonian routing protocol on the star graph that is asymptotically faster than gate-based routing; [2] a lower bound on the time to realize the shift unitary using 1d power-law interactions which, perhaps surprisingly, can be much slower than the time for the conventional Lieb-Robinson light cone to spread across the whole system. The latter result shares interesting connections to the classification of 1d quantum cellular automata and symmetry-protected topological order.

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Hamiltonian simulation of lattice gauge theories (LGTs) is a non-perturbative method of numerically solving gauge theories that offers novel avenues for studying scattering processes in gauge theories. With the advent of quantum computers, Hamiltonian simulation of LGTs has become a reality. Simulating scattering on quantum computers requires the preparation of initial scattering states in the interacting theory on the quantum hardware. Current state preparation methods involve bridging the scattering states in the free theory to the ones in the interacting theory adiabatically. Such quantum algorithms have limitations when applied to LGTs, and they tend to be computational resource intensive, rendering their implementation a challenge on the noisy intermediate-scale quantum (NISQ) era devices. In this work, we propose a wave packet state preparation algorithm for a 1+1D Z2 LGT coupled to dynamical matter. We show how this algorithm circumvents the adiabatic process by building and implementing the wave packet creation operators directly in the interacting theory using an optimized ansatz consisting of hadronic degrees of freedom in the confined Z2 LGT. Moreover, we numerically confirm the validity of this ansatz for a U(1) LGT in 1+1D. Finally, we demonstrate the viability of our algorithm for NISQ devices by comparing the classical simulation with the results obtained using the Quantinuum H1-1 quantum computer upon a simple symmetry-based noise mitigation technique.

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Minimizing the energy of a many-body system is a fundamental problem in many fields. Although we hope a quantum computer can help us solve this problem better than classical computers, we have a very limited understanding of where a quantum advantage may be found. In this talk, I will present some recent theoretical advances that shed light on quantum advantages in this domain. First, I describe rigorous analyses of the Quantum Approximate Optimization Algorithm applied to minimizing energies of classical spin glasses. For certain families of spin glasses, we find the QAOA has a quantum advantage over the best known classical algorithms. Second, we study the problem of finding a local minimum of the energy of quantum systems. While local minima are much easier to find than ground states, we show that finding a local minimum under thermal perturbations is computationally hard for classical computers, but easy for quantum computers.

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As quantum technologies are expected to provide us with unprecedented benefits, identifying what quantum-mechanical properties are useful is a pivotal question. Quantum resource theories provide a unified framework to analyze such quantum properties, which has been successful in the understanding of fundamental properties such as entanglement and coherence. While these are examples of convex resources, for which quantum advantages can always be identified, many physical resources are described by a non-convex set of free states and their interpretation has so far remained elusive. In this work, we address the fundamental question of the usefulness of quantum resources without convexity assumption, by providing two operational interpretations of the generalized robustness resource measure in general resource theories. On the one hand, we characterize the generalized robustness in terms of a non-linear resource witness and reveal that any state is more advantageous than a free one in some multi-copy channel discrimination task. On the other hand, we consider a scenario where a theory is characterized by multiple constraints and show that the generalized robustness coincides with the worst-case advantage in a single-copy channel discrimination setting. We further extend these results to the weight resource measure and QRTs for quantum channels and quantum instruments. Based on these characterizations, we conclude that every quantum resource state shows a qualitative and quantitative advantage in discrimination problems in a general resource theory even without any assumption on the structure of the free sets. This talk is based on arXiv:2310.09154 and arXiv:2310.09321.

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