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Relational quantum computing using only maximally mixed initial qubit states

Terry Rudolph, S. Virmani

Published Jul 7, 2021• 1 citation
OpenAlex

Abstract

We disprove the conjecture of [1], namely that it would require smarter authors to find a way of making the two-qubit singlet/triplet measurement quantum computationally universal given an ensemble of initial single qubit states with less than three linearly independent Bloch vectors. We show, in fact, that an initial ensemble of maximally mixed single qubits suffices.

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The two-qubit singlet/triplet measurement is universal for quantum computing given only maximally-mixed initial states

Terry Rudolph, S. Virmani

Abstract In order to delineate which minimalistic physical primitives can enable the full power of universal quantum computing, it has been fruitful to consider various measurement based architectures which reduce or eliminate the use of coherent unitary evolution, and also involve operations that are physically natural. In this context previous works had shown that the triplet-singlet measurement of two qubit angular momentum (or equivalently two qubit exchange symmetry) yields the power of quantum computation given access to a few additional different single qubit states or gates. However, Freedman, Hastings and Shokrian-Zini1 recently proposed a remarkable conjecture, called the ‘STP=BQP’ conjecture, which states that the two-qubit singlet/triplet measurement is quantum computationally universal given only an initial ensemble of maximally mixed single qubits. In this work we prove this conjecture. This provides a method for quantum computing that is fully rotationally symmetric (i.e. reference frame independent), using primitives that are physically very-accessible, naturally resilient to certain forms of error, and provably the simplest possible.

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We consider quantum computing in the one-qubit model where the starting state of a quantum computer consists of k qubits in a pure state and n − k qubits in a maximally mixed state. We ask the following question: is there a general method for simulating an arbitrary m-qubit pure state quantum computation by a quantum computation in the k-qubit model? We show that, under certain constraints, this is impossible, unless m = O(k + log n). 1.

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Qubit vs Observable Resource Trade-Offs in Measurement-Based Quantum Computation

Simon Perdrix

Since quantum measurement is universal for quantum computation (Nielsen [3]), the minimization of the resources required for measurement‐based quantum computation is a crucial point ([3, 1, 4, 5, 6]). We have shown in [6] that a set of observables composed of one two‐qubit measurement and three one‐qubit measurements form a universal set for quantum computation if one ancillary qubit is available as an additional resource. This set is minimal in terms of two‐qubit measurements, because at least one multi‐qubit operation is needed to create entanglement, and is also minimal in terms of ancillary qubits, because at least one ancillary qubit is required to simulate reversible transformations with projective measurements only. We discuss in this paper the required number of one‐qubit measurements for a set of observables to be universal. A set composed of one two‐qubit measurement and only two one‐qubit measurements is proved universal, at the cost of a second ancillary qubit. Thus an additional ancillary qubit seems to be required to obtain a smaller universal set of measurements. One may conclude to a trade off between ancillary qubits and observables in measurement‐based quantum computation.

AIP conference proceedings•Jan 1, 2004•2 citations
We disprove the conjecture of [1], namely that it would require smarter authors to find a way of making the two-qubit singlet/triplet measurement quantum computationally universal given an ensemble of initial single qubit states with less than three linearly independent Bloch vectors. We show, in fact, that an initial ensemble of maximally mixed single qubits suffices.
Abstract In order to delineate which minimalistic physical primitives can enable the full power of universal quantum computing, it has been fruitful to consider various measurement based architectures which reduce or eliminate the use of coherent unitary evolution, and also involve operations that are physically natural. In this context previous works had shown that the triplet-singlet measurement of two qubit angular momentum (or equivalently two qubit exchange symmetry) yields the power of quantum computation given access to a few additional different single qubit states or gates. However, Freedman, Hastings and Shokrian-Zini1 recently proposed a remarkable conjecture, called the ‘STP=BQP’ conjecture, which states that the two-qubit singlet/triplet measurement is quantum computationally universal given only an initial ensemble of maximally mixed single qubits. In this work we prove this conjecture. This provides a method for quantum computing that is fully rotationally symmetric (i.e. reference frame independent), using primitives that are physically very-accessible, naturally resilient to certain forms of error, and provably the simplest possible.
We prove that universal quantum computation is possible using only (i) the physically natural measurement on two qubits which distinguishes the singlet from the triplet subspace and (ii) qubits prepared in almost any three different (potentially highly mixed) states. In some sense this measurement is a 'more universal' dynamical element than a universal two-qubit unitary gate, since the latter must be supplemented by measurement. Because of the rotational invariance of the measurement used, our scheme is robust to collective decoherence in a manner very different to previous proposals—in effect it is only ever sensitive to the relational properties of the qubits.
In standard quantum computation, the initial state is pure and the answer is determined by making a measurement of some of the bits in the computational basis. What can be accomplished if the initial state is a highly mixed state and the answer is determined by measuring the expectation of ${\ensuremath{\sigma}}_{z}$ on the first bit with bounded sensitivity? This is the situation in high temperature ensemble quantum computation. We show that in this model it is possible to perform interesting physics simulations that have no known efficient classical algorithms, even though the model is less powerful than standard quantum computation in the presence of oracles.
The coupling of two spin-1/2 systems is a canonical example of composite-state construction, angular-momentum addition, exchange symmetry, and quantum entanglement. This article develops a reproducible computational treatment of the problem using two complementary frameworks. MATLAB is used to implement the conventional matrix-mechanics description through vectors, Kronecker products, Hermitian operators, eigendecomposition, density matrices, and partial traces. PennyLane is used to implement the same states using quantum gates, QNodes, state-vector simulation, observable expectation values, finite-shot sampling, and reduced-density-matrix measurements. The singlet and triplet sectors are derived analytically and verified numerically to be eigenstates of the total-spin operators S^2 and Sz. Fidelity is used to compare the state vectors of MATLAB and PennyLane while accounting for the physically irrelevant global phase. Single-particle Bloch vectors and reduced density matrices are employed to show that the entangled states have maximally mixed local states, so their individual Bloch vectors vanish even though the global two-qubit states are pure. A basis-change circuit is further introduced to distinguish the relative phase of the singlet and m = 0 triplet states, which cannot be resolved by computational-basis populations alone. The study provides a bridge between textbook angular-momentum theory, classical scientific computing, and modern gate-based quantum programming.
Since quantum measurement is universal for quantum computation (Nielsen [3]), the minimization of the resources required for measurement‐based quantum computation is a crucial point ([3, 1, 4, 5, 6]). We have shown in [6] that a set of observables composed of one two‐qubit measurement and three one‐qubit measurements form a universal set for quantum computation if one ancillary qubit is available as an additional resource. This set is minimal in terms of two‐qubit measurements, because at least one multi‐qubit operation is needed to create entanglement, and is also minimal in terms of ancillary qubits, because at least one ancillary qubit is required to simulate reversible transformations with projective measurements only. We discuss in this paper the required number of one‐qubit measurements for a set of observables to be universal. A set composed of one two‐qubit measurement and only two one‐qubit measurements is proved universal, at the cost of a second ancillary qubit. Thus an additional ancillary qubit seems to be required to obtain a smaller universal set of measurements. One may conclude to a trade off between ancillary qubits and observables in measurement‐based quantum computation.