The think tank has produced a report, written by economist Michael Johnson, which calls for today's tax relief framework to be swept away and replaced with a simpler, more redistributive 50p per £1 saved.
This should be paid irrespective of the saver's taxpaying status, Mr Johnson said.
The paper, called Incentivising retirement saving: the end of tax relief, and a new beginning, also proposed a combined contributions limit for pensions and ISAs.
The CPS suggested a hard cash savings incentive of up to £4,000 per year.
It argued that current retirement saving incentives, estimated to cost £54 billion last year, were "an ineffective, and inequitable, use of Treasury funds".
The think tank said these have "failed to catalyse the broad-based savings culture that Britain needs".
Mr Johnson hopes his proposals will appeal to both the political left and right.

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The eight specific proposals are:

• Pension contributions from employers should be treated as part of employees' gross income, and taxed as such.
• Tax relief on pension contributions should be replaced by a Treasury contribution of 50p per £1 saved, up to an annual allowance, paid irrespective of the saver's taxpaying status.
• ISA and pension products should share an annual combined contribution limit of £30,000, available for saving within ISA or pension products (or any combination thereof). This would replace the current ISA and pensions tax-advantaged allowances.
• The 25% tax-free lump sum should be scrapped, with accrued rights to it protected.
• The Lifetime Allowance should be scrapped. It adds considerably complexity to the pensions landscape, and with a £30,000 combined contributions limit for pensions and ISAs, it would become less relevant over time.
• The 10p tax rebate on pension assets' dividend income should be reinstated.
• People should be able to bequeath unused pension pot assets to third parties free of Inheritance Tax (perhaps limited to £100,000), provided that the assets remained within a pensions framework.
• The annual allowance should be set at £8,000, with prior years' unutilised allowances being permitted to be rolled up, perhaps over as much as ten years, all subject to modelling confirmation.