[Paper Review] Computing with highly mixed states

Andris Ambainis, Leonard J. Schulman+1 more|ArXiv.org|Mar 29, 2000

Quantum Computing Algorithms and Architecture6 references15 citations

TL;DR

This paper investigates quantum computation in a model where only k qubits are in a pure state while the remaining n−k are maximally mixed. It proves that simulating arbitrary m-qubit quantum computations in this model is impossible unless m = O(k + log n), establishing a fundamental limit on computational power in mixed-state quantum computing under realistic initialization constraints.

ABSTRACT

We consider quantum computing in the k-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).

Motivation & Objective

• To determine whether arbitrary quantum computations can be simulated in a one-qubit model where only k qubits are in a pure state and the rest are maximally mixed.
• To assess the feasibility of scalable quantum computation in liquid NMR systems, where initialization produces a binomial distribution over pure states with low polarization.
• To investigate whether a small number of initialized (clean) qubits can enable universal quantum computation in the presence of many maximally mixed qubits.
• To establish theoretical limits on the computational power of such mixed-state quantum computing models.
• To explore the connection between symmetric group representations and the distinguishability of quantum states in this model.

Proposed method

• Analyzes the one-qubit model where k qubits are in a pure state and n−k in a maximally mixed state, treating the initial state as fixed and the only input as the sequence of quantum operations.
• Uses representation theory of the symmetric group to analyze the invariance properties of quantum states under permutations of qubits.
• Applies dimensionality bounds on subspaces invariant under symmetric group actions to derive limits on state distinguishability.
• Employs a key inequality (Lemma 7) relating the dimension of intersections of subspaces under group actions to the overall dimension and number of irreducible representations.
• Uses the fact that unitary operations preserving the symmetric group structure constrain how much entanglement or superposition can be created.
• Applies a probabilistic argument based on the trace distance and distinguishability of encoded states to show that only logarithmic-sized computations can be reliably simulated.

Experimental results

Research questions

• RQ1 Can arbitrary m-qubit quantum computations be simulated in a model where only k qubits are in a pure state and the rest are maximally mixed?
• RQ2 What is the maximum number of qubits m for which such a simulation is possible, given k clean qubits and n−k maximally mixed qubits?
• RQ3 Why is the one-clean-qubit model (k=1) insufficient for universal quantum computation beyond O(log n) qubits?
• RQ4 How do the symmetries of the symmetric group constrain the distinguishability of quantum states in this mixed-state model?
• RQ5 Can the 3-qubit model with one clean qubit simulate all of NC1, and if so, why does this fail to extend to QNC1?

Key findings

• It is impossible to simulate arbitrary m-qubit quantum computations in the k-qubit pure state model unless m = O(k + log n).
• The simulation power is fundamentally limited by the number of clean qubits and the logarithmic scaling with n, making it no more efficient than classical exhaustive search for large m.
• The key result is derived using representation theory of the symmetric group, particularly bounding the dimension of invariant subspaces under group actions.
• Lemma 7 establishes a quantitative bound: |dim(X ∩ W_t) − dim(X ∩ Y ∩ W_t)| ≤ √(4cn/M) ⋅ dim W_t, which is central to the proof.
• The distinguishability of encoded states decreases exponentially with m, making reliable computation impossible beyond O(k + log n) qubits.
• Despite the limitation, a 3-qubit register with one clean qubit can simulate all languages in NC1, demonstrating a non-trivial computational capability within the model.