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Register a new userQuantum circuit synthesis for generalized coherent states
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Authors
Rolando D. Somma
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We present a method that outputs a sequence of simple unitary operations to prepare a given quantum state that is a generalized coherent state. Our method takes as inputs the expectation values of some relevant observables on the state to be prepared. Such expectation values can be estimated by performing projective measurements on $O(M^3 log(M/delta)/epsilon^2)$ copies of the state, where $M$ is the dimension of an associated Lie algebra, $epsilon$ is a precision parameter, and $1-delta$ is the required confidence level. The method can be implemented on a classical computer and runs in time $O(M^4 log(M/epsilon))$. It provides $O(M log(M/epsilon))$ simple unitaries that form the sequence. The number of all computational resources is then polynomial in $M$, making the whole procedure very efficient in those cases where $M$ is significantly smaller than the Hilbert space dimension. When the algebra of relevant observables is determined by some Pauli matrices, each simple unitary may be easily decomposed into two-qubit gates. We discuss applications to quantum state tomography and classical simulations of quantum circuits.
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Quantum circuit synthesis is the process in which an arbitrary unitary operation is decomposed into a sequence of gates from a universal set, typically one which a quantum computer can implement both efficiently and fault-tolerantly. As physical implementations of quantum computers improve, the need is growing for tools which can effectively synthesize components of the circuits and algorithms they will run. Existing algorithms for exact, multi-qubit circuit synthesis scale exponentially in the number of qubits and circuit depth, leaving synthesis intractable for circuits on more than a handful of qubits. Even modest improvements in circuit synthesis procedures may lead to significant advances, pushing forward the boundaries of not only the size of solvable circuit synthesis problems, but also in what can be realized physically as a result of having more efficient circuits. We present a method for quantum circuit synthesis using deterministic walks. Also termed pseudorandom walks, these are walks in which once a starting point is chosen, its path is completely determined. We apply our method to construct a parallel framework for circuit synthesis, and implement one such version performing optimal $T$-count synthesis over the Clifford+$T$ gate set. We use our software to present examples where parallelization offers a significant speedup on the runtime, as well as directly confirm that the 4-qubit 1-bit full adder has optimal $T$-count 7 and $T$-depth 3.
Generalized coherent states are developed for SU(n) systems for arbitrary $n$. This is done by first iteratively determining explicit representations for the SU(n) coherent states, and then determining parametric representations useful for applications. For SU(n), the set of coherent states is isomorphic to a coset space $SU(n)/SU(n-1)$, and thus shows the geometrical structure of the coset space. These results provide a convenient $(2n - 1)$--dimensional space for the description of arbitrary SU(n) systems. We further obtain the metric and measure on the coset space, and show some properties of the SU(n) coherent states.
Variational quantum algorithms that are used for quantum machine learning rely on the ability to automatically differentiate parametrized quantum circuits with respect to underlying parameters. Here, we propose the rules for differentiating quantum circuits (unitaries) with arbitrary generators. Unlike the standard parameter shift rule valid for unitaries generated by operators with spectra limited to at most two unique eigenvalues (represented by involutory and idempotent operators), our approach also works for generators with a generic non-degenerate spectrum. Based on a spectral decomposition, we derive a simple recipe that allows explicit derivative evaluation. The derivative corresponds to the weighted sum of measured expectations for circuits with shifted parameters. The number of function evaluations is equal to the number of unique positive non-zero spectral gaps (eigenvalue differences) for the generator. We apply the approach to relevant examples of two-qubit gates, among others showing that the fSim gate can be differentiated using four measurements. Additionally, we present generalized differentiation rules for the case of Pauli string generators, based on distinct shifts (here named as the triangulation approach), and analyse the variance for derivative measurements in different scenarios. Our work offers a toolbox for the efficient hardware-oriented differentiation needed for circuit optimization and operator-based derivative representation.
The current phase of quantum computing is in the Noisy Intermediate-Scale Quantum (NISQ) era. On NISQ devices, two-qubit gates such as CNOTs are much noisier than single-qubit gates, so it is essential to minimize their count. Quantum circuit synthesis is a process of decomposing an arbitrary unitary into a sequence of quantum gates, and can be used as an optimization tool to produce shorter circuits to improve overall circuit fidelity. However, the time-to-solution of synthesis grows exponentially with the number of qubits. As a result, synthesis is intractable for circuits on a large qubit scale. In this paper, we propose a hierarchical, block-by-block optimization framework, QGo, for quantum circuit optimization. Our approach allows an exponential cost optimization to scale to large circuits. QGo uses a combination of partitioning and synthesis: 1) partition the circuit into a sequence of independent circuit blocks; 2) re-generate and optimize each block using quantum synthesis; and 3) re-compose the final circuit by stitching all the blocks together. We perform our analysis and show the fidelity improvements in three different regimes: small-size circuits on real devices, medium-size circuits on noise simulations, and large-size circuits on analytical models. Using a set of NISQ benchmarks, we show that QGo can reduce the number of CNOT gates by 29.9% on average and up to 50% when compared with industrial compilers such as t|ket>. When executed on the IBM Athens system, shorter depth leads to higher circuit fidelity. We also demonstrate the scalability of our QGo technique to optimize circuits of 60+ qubits. Our technique is the first demonstration of successfully employing and scaling synthesis in the compilation toolchain for large circuits. Overall, our approach is robust for direct incorporation in production compiler toolchains.
Quantum Computing has been evolving in the last years. Although nowadays quantum algorithms performance has shown superior to their classical counterparts, quantum decoherence and additional auxiliary qubits needed for error tolerance routines have been huge barriers for quantum algorithms efficient use. These restrictions lead us to search for ways to minimize algorithms costs, i.e the number of quantum logical gates and the depth of the circuit. For this, quantum circuit synthesis and quantum circuit optimization techniques are explored. We studied the viability of using Projective Simulation, a reinforcement learning technique, to tackle the problem of quantum circuit synthesis for noise quantum computers with limited number of qubits. The agent had the task of creating quantum circuits up to 5 qubits to generate GHZ states in the IBM Tenerife (IBM QX4) quantum processor. Our simulations demonstrated that the agent had a good performance but its capacity for learning new circuits decreased as the number of qubits increased.
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